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<front>
<journal-meta>
<journal-id journal-id-type="publisher-id">Front. Physiol.</journal-id>
<journal-title-group>
<journal-title>Frontiers in Physiology</journal-title>
<abbrev-journal-title abbrev-type="pubmed">Front. Physiol.</abbrev-journal-title>
</journal-title-group>
<issn pub-type="epub">1664-042X</issn>
<publisher>
<publisher-name>Frontiers Media S.A.</publisher-name>
</publisher>
</journal-meta>
<article-meta>
<article-id pub-id-type="publisher-id">1731863</article-id>
<article-id pub-id-type="doi">10.3389/fphys.2025.1731863</article-id>
<article-version article-version-type="Version of Record" vocab="NISO-RP-8-2008"/>
<article-categories>
<subj-group subj-group-type="heading">
<subject>Original Research</subject>
</subj-group>
</article-categories>
<title-group>
<article-title>Structural integrity of RyR2 clusters controls cardiac calcium leak</article-title>
<alt-title alt-title-type="left-running-head">Noren and Shiferaw</alt-title>
<alt-title alt-title-type="right-running-head">
<ext-link ext-link-type="uri" xlink:href="https://doi.org/10.3389/fphys.2025.1731863">10.3389/fphys.2025.1731863</ext-link>
</alt-title>
</title-group>
<contrib-group>
<contrib contrib-type="author">
<name>
<surname>Noren</surname>
<given-names>Andrew</given-names>
</name>
<xref ref-type="aff" rid="aff1"/>
<role vocab="credit" vocab-identifier="https://credit.niso.org/" vocab-term="Conceptualization" vocab-term-identifier="https://credit.niso.org/contributor-roles/conceptualization/">Conceptualization</role>
<role vocab="credit" vocab-identifier="https://credit.niso.org/" vocab-term="Data curation" vocab-term-identifier="https://credit.niso.org/contributor-roles/data-curation/">Data curation</role>
<role vocab="credit" vocab-identifier="https://credit.niso.org/" vocab-term="Formal analysis" vocab-term-identifier="https://credit.niso.org/contributor-roles/formal-analysis/">Formal Analysis</role>
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<role vocab="credit" vocab-identifier="https://credit.niso.org/" vocab-term="Visualization" vocab-term-identifier="https://credit.niso.org/contributor-roles/visualization/">Visualization</role>
<role vocab="credit" vocab-identifier="https://credit.niso.org/" vocab-term="Writing &#x2013; original draft" vocab-term-identifier="https://credit.niso.org/contributor-roles/writing-original-draft/">Writing - original draft</role>
<role vocab="credit" vocab-identifier="https://credit.niso.org/" vocab-term="Writing &#x2013; review &#x26; editing" vocab-term-identifier="https://credit.niso.org/contributor-roles/Writing - review &#x26; editing/">Writing - review and editing</role>
</contrib>
<contrib contrib-type="author" corresp="yes">
<name>
<surname>Shiferaw</surname>
<given-names>Yohannes</given-names>
</name>
<xref ref-type="aff" rid="aff1"/>
<xref ref-type="corresp" rid="c001">&#x2a;</xref>
<uri xlink:href="https://loop.frontiersin.org/people/22916"/>
<role vocab="credit" vocab-identifier="https://credit.niso.org/" vocab-term="Conceptualization" vocab-term-identifier="https://credit.niso.org/contributor-roles/conceptualization/">Conceptualization</role>
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</contrib-group>
<aff id="aff1">
<institution>Department of Physics and Astronomy, California State University</institution>, <city>Los Angeles</city>, <state>CA</state>, <country country="US">United States</country>
</aff>
<author-notes>
<corresp id="c001">
<label>&#x2a;</label>Correspondence: Yohannes Shiferaw, <email xlink:href="mailto:yshiferaw@csun.edu">yshiferaw@csun.edu</email>
</corresp>
</author-notes>
<pub-date publication-format="electronic" date-type="pub" iso-8601-date="2025-12-09">
<day>09</day>
<month>12</month>
<year>2025</year>
</pub-date>
<pub-date publication-format="electronic" date-type="collection">
<year>2025</year>
</pub-date>
<volume>16</volume>
<elocation-id>1731863</elocation-id>
<history>
<date date-type="received">
<day>24</day>
<month>10</month>
<year>2025</year>
</date>
<date date-type="rev-recd">
<day>23</day>
<month>11</month>
<year>2025</year>
</date>
<date date-type="accepted">
<day>25</day>
<month>11</month>
<year>2025</year>
</date>
</history>
<permissions>
<copyright-statement>Copyright &#xa9; 2025 Noren and Shiferaw.</copyright-statement>
<copyright-year>2025</copyright-year>
<copyright-holder>Noren and Shiferaw</copyright-holder>
<license>
<ali:license_ref start_date="2025-12-09">https://creativecommons.org/licenses/by/4.0/</ali:license_ref>
<license-p>This is an open-access article distributed under the terms of the <ext-link ext-link-type="uri" xlink:href="https://creativecommons.org/licenses/by/4.0/">Creative Commons Attribution License (CC BY)</ext-link>. The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.</license-p>
</license>
</permissions>
<abstract>
<sec>
<title>Background</title>
<p>Calcium (Ca) leak from the sarcoplasmic reticulum contributes to cardiac arrhythmias, yet the structural mechanisms regulating spontaneous Ca release from ryanodine receptor type 2 (RyR2) clusters remain poorly understood.</p>
</sec>
<sec>
<title>Methods</title>
<p>We developed a computational model in which each RyR2 channel comprises four interacting subunits embedded within spatially organized clusters. This framework captures both cooperative gating within individual channels and coupling between neighboring channels.</p>
</sec>
<sec>
<title>Results</title>
<p>Our simulations reveal that spontaneous Ca spark timing depends exponentially on RyR2 cluster structural integrity. This exponential sensitivity means that modest disruptions in cluster structure, such as partial fragmentation, can increase spontaneous Ca spark frequency by 100&#x2013;1,000 fold.</p>
</sec>
<sec>
<title>Conclusions</title>
<p>Cluster structural integrity provides a powerful control mechanism for Ca leak and represents a promising therapeutic target for restoring Ca homeostasis in cardiac myocytes.</p>
</sec>
</abstract>
<kwd-group>
<kwd>RyR2</kwd>
<kwd>calcium release channel</kwd>
<kwd>cardiac arhythmias</kwd>
<kwd>channel cluster</kwd>
<kwd>spontaneous release</kwd>
</kwd-group>
<funding-group>
<funding-statement>The authors declare that financial support was received for the research and/or publication of this article. This work was supported by the National Institute of General Medical Sciences (Award Number: 1R16GM153647-01 to YS) and the National Science Foundation (Award Number: 2320846 to YS). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.</funding-statement>
</funding-group>
<counts>
<fig-count count="6"/>
<table-count count="0"/>
<equation-count count="4"/>
<ref-count count="40"/>
<page-count count="11"/>
</counts>
<custom-meta-group>
<custom-meta>
<meta-name>section-in-acceptance</meta-name>
<meta-value>Cell Physiology</meta-value>
</custom-meta>
</custom-meta-group>
</article-meta>
</front>
<body>
<sec sec-type="intro" id="s1">
<title>Introduction</title>
<p>Calcium (Ca) signaling is fundamental to cardiac muscle contraction, with tightly regulated intracellular Ca release essential for effective excitation-contraction coupling (<xref ref-type="bibr" rid="B5">Bers, 2002</xref>). The RyR2, a large Ca release channel located on the sarcoplasmic reticulum (SR) membrane, plays a central role in this process. RyR2 is activated through calcium-induced calcium release (CICR), in which a small influx of Ca via L-type Ca channels (LCCs) during the cardiac action potential triggers a much larger release of Ca from the SR. Recent advances in super-resolution imaging techniques, have provided unprecedented insight into the spatial organization of RyR2 within cardiomyocytes (<xref ref-type="bibr" rid="B31">Sheard et al., 2022</xref>; <xref ref-type="bibr" rid="B18">Hurley et al., 2023</xref>; <xref ref-type="bibr" rid="B27">Mesa et al., 2022</xref>; <xref ref-type="bibr" rid="B3">Baddeley et al., 2009</xref>; <xref ref-type="bibr" rid="B21">Kolstad et al., 2018</xref>; <xref ref-type="bibr" rid="B33">Soeller and Jayasinghe, 2018</xref>; <xref ref-type="bibr" rid="B20">Jayasinghe et al., 2009</xref>) These methods have revealed that RyR2 channels form distinct, heterogeneous clusters whose size, density, and arrangement are critical determinants of Ca signaling dynamics. This detailed structural information now provides a framework for investigating how the organization of RyR2 clusters contributes to normal cardiac physiology, and how its disruption may lead to arrhythmogenesis (<xref ref-type="bibr" rid="B11">Fowler and Zissimopoulos, 2022</xref>; <xref ref-type="bibr" rid="B35">Waddell et al., 2023</xref>; <xref ref-type="bibr" rid="B25">Macquaide et al., 2015</xref>; <xref ref-type="bibr" rid="B10">Dixon, 2022</xref>; <xref ref-type="bibr" rid="B4">Benitah et al., 2021</xref>; <xref ref-type="bibr" rid="B9">Chen-Izu et al., 2007</xref>).</p>
<p>New imaging techniques have revealed that RyR2 clusters are not static structures, but instead show dynamic organization that can change significantly in disease (<xref ref-type="bibr" rid="B21">Kolstad et al., 2018</xref>; <xref ref-type="bibr" rid="B35">Waddell et al., 2023</xref>; <xref ref-type="bibr" rid="B16">Hiess et al., 2018</xref>). In healthy cardiomyocytes, RyR2 channels are arranged in compact organized clusters at dyads. In disease, however, this organization becomes disrupted. A common observation is cluster fragmentation, where large RyR2 assemblies break into smaller, irregularly shaped groupings. This reduces the number of channels per cluster and alters their spatial arrangement, with important consequences for Ca signaling. In an elegant study <xref ref-type="bibr" rid="B31">Sheard et al. (2022)</xref> used enhanced expansion microscopy to visualize this remodeling in three dimensions, revealing that fragmented RyR2 clusters often exhibit a frayed appearance. In these clusters, RyR2 channels remain partially anchored near the center of the cluster, where a structural protein called junctophilin-2 (JPH2) helps tether them to the membrane, but many RyRs extend outward in a disorganized, loosely connected pattern. This &#x201c;fraying&#x201d; suggests a loss of structural integrity in the regions where JPH2 is absent, marking the early breakdown of the tightly organized release sites essential for normal Ca signaling. In heart failure, similar changes are observed. Super-resolution imaging studies, such as those by <xref ref-type="bibr" rid="B21">Kolstad et al. (2018)</xref>, have shown that RyR2 clusters become dispersed, forming smaller and more loosely organized sub-clusters. Also, in persistent atrial fibrillation, <xref ref-type="bibr" rid="B25">Macquaide et al. (2015)</xref> demonstrated that RyR2 cluster fragmentation results in more numerous, closely spaced clusters within Ca release units, accompanied by a greater than 50% increase in spontaneous Ca spark frequency. These structural changes are closely linked to key functional defects in heart failure, including elevated diastolic Ca leak, reduced contractile strength, and a greater risk of arrhythmias (<xref ref-type="bibr" rid="B4">Benitah et al., 2021</xref>; <xref ref-type="bibr" rid="B1">Ai et al., 2005</xref>; <xref ref-type="bibr" rid="B26">Marx and Marks, 2013</xref>).</p>
<p>Computational models have provided important insights into how RyR2 spatial organization affects calcium signaling. Recent studies have demonstrated that RyR2 interactions via Ca diffusion are critical for coordinating calcium release (<xref ref-type="bibr" rid="B8">Cannell et al., 2013</xref>), with network connectivity affecting whole-cell calcium oscillations (<xref ref-type="bibr" rid="B13">Gao et al., 2023</xref>) and cluster geometry influencing calcium spark properties (<xref ref-type="bibr" rid="B19">Iaparov et al., 2021</xref>; <xref ref-type="bibr" rid="B24">Li et al., 2025</xref>). These findings establish that spatial effects operate through multiple mechanisms, particularly diffusive calcium coupling between channels. However, existing models have not incorporated direct subunit coupling between adjacent RyR2 channels within clusters. Such coupling arises from the physical contacts between neighboring channels revealed by cryo-EM studies (<xref ref-type="bibr" rid="B7">Cabra et al., 2016</xref>) and represents a distinct mechanism through which cluster connectivity may regulate calcium release. In this study, we model a cluster of RyR2 channels as an array of interacting tetramers. Each RyR2 channel is composed of four subunits, and channels within the cluster are arranged with specific geometrical contacts that reflect their structural organization. We analyze the stochastic dynamics of this system to determine how the arrangement of neighboring channels influences the behavior of the cluster. Our focus is to understand how cluster architecture controls the timing of spontaneous Ca sparks. A central finding is that, at resting Ca concentrations, the frequency of spontaneous Ca sparks is exponentially sensitive to the structural integrity of the cluster. In this regime, small changes in arrangement, such as increased spacing, reduced connectivity, or fragmentation into sub-clusters, can increase the frequency of spontaneous Ca sparks by several orders of magnitude. This result highlights the crucial role of intact cluster architecture in maintaining coordinated channel closure at rest. We further show that frayed clusters are particularly vulnerable, since small peripheral groups of RyR2s can recruit larger assemblies through diffusive Ca coupling. Together, these results identify cluster architecture as a dominant control mechanism for Ca leak under diastolic conditions, linking cluster integrity to pathological Ca handling.</p>
</sec>
<sec sec-type="methods" id="s2">
<title>Methods</title>
<sec id="s2-1">
<title>Computational model of RyR2 tetramer</title>
<p>The RyR2 channel is a tetramer composed of four identical subunits that gate a shared central pore. Structural studies show that these subunits are tightly packed and physically interact at their interfaces (<xref ref-type="bibr" rid="B7">Cabra et al., 2016</xref>; <xref ref-type="bibr" rid="B38">Woll and Van Petegem, 2022</xref>). Motivated by these observations and by our earlier modeling work (<xref ref-type="bibr" rid="B15">Greene et al., 2023</xref>), we describe a simplified RyR2 model in which each subunit can exist in one of two conformational states. A central assumption of the model is that adjacent subunits interact energetically, such that conformational mismatches are penalized. In particular, if a subunit transitions from closed to open while its neighbors remain closed, the resulting mismatch produces an energetic cost. This is implemented by modifying the transition rate so that the opening rate of a closed subunit is reduced by a multiplicative factor when neighbors are closed and increased when neighbors are open. An additional assumption is that a channel is considered functionally open when three or more subunits are in the open state. This requirement was established in our previous study (<xref ref-type="bibr" rid="B15">Greene et al., 2023</xref>), where we demonstrated that RyR2 channels must exhibit cooperative gating to remain reliably shut at diastolic Ca concentrations. Without this constraint, channels cannot maintain stable closure during rest.</p>
<p>To apply this model, we represent the RyR2 channel as four subunits labeled <inline-formula id="inf1">
<mml:math id="m1">
<mml:mrow>
<mml:msub>
<mml:mi>s</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
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</inline-formula>, with <inline-formula id="inf2">
<mml:math id="m2">
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
<mml:mo>,</mml:mo>
<mml:mn>2</mml:mn>
<mml:mo>,</mml:mo>
<mml:mn>3</mml:mn>
<mml:mo>,</mml:mo>
<mml:mn>4</mml:mn>
<mml:mo>,</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> and periodic boundary conditions. Each subunit can be in one of two states: closed <inline-formula id="inf3">
<mml:math id="m3">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:msub>
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<mml:mo>&#x3d;</mml:mo>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
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</mml:mrow>
</mml:math>
</inline-formula> or open <inline-formula id="inf4">
<mml:math id="m4">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:msub>
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<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula>. This configuration ensures that every subunit interacts with its two immediate neighbors, consistent with the geometry of the RyR2 channel. Transitions between subunit states are governed by rates that depend on the conformational states of neighboring subunits. The forward (opening) and backward (closing) transition rates for subunit <inline-formula id="inf5">
<mml:math id="m5">
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> are given by<disp-formula id="e1">
<mml:math id="m6">
<mml:mrow>
<mml:msub>
<mml:mi>k</mml:mi>
<mml:mi>i</mml:mi>
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<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
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<mml:mi>i</mml:mi>
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<mml:msub>
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<mml:mi>i</mml:mi>
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<mml:mfenced open="(" close=")" separators="&#x7c;">
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</mml:msubsup>
<mml:mo>,</mml:mo>
</mml:mrow>
</mml:math>
<label>(1)</label>
</disp-formula>where <inline-formula id="inf6">
<mml:math id="m7">
<mml:mrow>
<mml:msub>
<mml:mi>k</mml:mi>
<mml:mi>f</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> and <inline-formula id="inf7">
<mml:math id="m8">
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<mml:mi>k</mml:mi>
<mml:mi>b</mml:mi>
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</inline-formula> are the intrinsic forward and backward rates in the absence of coupling. The factor <inline-formula id="inf8">
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</mml:mrow>
</mml:math>
</inline-formula> captures the influence of neighboring subunits and is defined as:<disp-formula id="e2">
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</mml:mrow>
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<mml:mo>.</mml:mo>
</mml:mrow>
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<label>(2)</label>
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</p>
<p>Here, <inline-formula id="inf9">
<mml:math id="m11">
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<mml:mi>&#x3b2;</mml:mi>
<mml:mi>J</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> is a dimensionless parameter representing the strength of coupling between channels, and where <inline-formula id="inf10">
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<mml:mi>J</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> is the energy penalty. Also, <inline-formula id="inf11">
<mml:math id="m13">
<mml:mrow>
<mml:mi>&#x3b2;</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
<mml:mo>/</mml:mo>
<mml:msub>
<mml:mi>k</mml:mi>
<mml:mi>B</mml:mi>
</mml:msub>
<mml:mi>T</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> where <inline-formula id="inf12">
<mml:math id="m14">
<mml:mrow>
<mml:msub>
<mml:mi>k</mml:mi>
<mml:mi>B</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is Boltzman&#x2019;s constant, and <inline-formula id="inf13">
<mml:math id="m15">
<mml:mrow>
<mml:mi>T</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> is the temperature. These local interaction rules form the basis of a thermodynamically consistent model that accounts for cooperative interactions within individual RyR2 channels.</p>
</sec>
<sec id="s2-2">
<title>Computational model of an RyR2 cluster</title>
<p>Recent Cryo-EM imaging studies due to <xref ref-type="bibr" rid="B7">Cabra et al. (2016)</xref> reveals that RyR2 channels physically interact at their interfaces through specific molecular contacts between adjacent channels. Their high-resolution structural analysis identifies two distinct types of inter-channel arrangements: an &#x201c;adjoining&#x201d; configuration (<xref ref-type="fig" rid="F1">Figure 1A</xref>) and an &#x201c;oblique&#x201d; configuration (<xref ref-type="fig" rid="F1">Figure 1B</xref>). In the adjoining arrangement, channels are aligned in regular rows and columns, with horizontal neighbors coupling through their edge-to-edge contacts. In the oblique arrangement (<xref ref-type="fig" rid="F1">Figure 1B</xref>), channels adopt a staggered geometry with interactions between subunits 1&#x2013;3 and 2&#x2013;4 only. Cabra et al. observed that native RyR2 clusters typically form disordered structures that represent combinations of these two geometric arrangements rather than uniform arrays of a single type. To model these experimentally observed arrangements, we will consider cluster models based on both interaction types.</p>
<fig id="F1" position="float">
<label>FIGURE 1</label>
<caption>
<p>Architecture of RyR2 clusters. <bold>(A)</bold> <italic>Adjoining configuration.</italic> RyR2 tetramers are positioned in a side-by-side array, with neighboring channels aligned in rows and columns. In this arrangement, adjacent channels make contact along their lateral surfaces, consistent with the &#x201c;side-by-side&#x201d; geometry observed in cardiac cells. <bold>(B)</bold> <italic>Oblique configuration.</italic> RyR2 tetramers are connected in an alternating &#x201c;checkerboard&#x201d; pattern. This arrangement corresponds to the oblique interaction identified in structural studies, where subunits from diagonally offset channels (<xref ref-type="bibr" rid="B5">Bers, 2002</xref>; <xref ref-type="bibr" rid="B31">Sheard et al., 2022</xref>; <xref ref-type="bibr" rid="B18">Hurley et al., 2023</xref>; <xref ref-type="bibr" rid="B31">Sheard et al., 2022</xref>; <xref ref-type="bibr" rid="B18">Hurley et al., 2023</xref>; <xref ref-type="bibr" rid="B27">Mesa et al., 2022</xref>) form the interface.</p>
</caption>
<graphic xlink:href="fphys-16-1731863-g001.tif">
<alt-text content-type="machine-generated">Illustration depicting two configurations. On the left, labeled &#x22;A adjoining,&#x22; nine aligned grids each divided into four numbered sections: one, two, four, and three. On the right, labeled &#x22;B oblique,&#x22; nine similar grids rotated diagonally, also numbered one, two, four, and three. Each grid is divided with red lines.</alt-text>
</graphic>
</fig>
<p>To model heterogeneous arrays, we denote <inline-formula id="inf14">
<mml:math id="m16">
<mml:mrow>
<mml:msubsup>
<mml:mi>s</mml:mi>
<mml:mi>i</mml:mi>
<mml:mi>c</mml:mi>
</mml:msubsup>
</mml:mrow>
</mml:math>
</inline-formula> as the state of subunit <inline-formula id="inf15">
<mml:math id="m17">
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> in channel <inline-formula id="inf16">
<mml:math id="m18">
<mml:mrow>
<mml:mi>c</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>. Similarly we will denote <inline-formula id="inf17">
<mml:math id="m19">
<mml:mrow>
<mml:msubsup>
<mml:mi>s</mml:mi>
<mml:mi>j</mml:mi>
<mml:msup>
<mml:mi>c</mml:mi>
<mml:mo>&#x2032;</mml:mo>
</mml:msup>
</mml:msubsup>
</mml:mrow>
</mml:math>
</inline-formula> and <inline-formula id="inf18">
<mml:math id="m20">
<mml:mrow>
<mml:msubsup>
<mml:mi>s</mml:mi>
<mml:mi>k</mml:mi>
<mml:msup>
<mml:mi>c</mml:mi>
<mml:mo>&#x2033;</mml:mo>
</mml:msup>
</mml:msubsup>
</mml:mrow>
</mml:math>
</inline-formula> to be the state of subunits <inline-formula id="inf19">
<mml:math id="m21">
<mml:mrow>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> and <inline-formula id="inf20">
<mml:math id="m22">
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> in neighboring channels <inline-formula id="inf21">
<mml:math id="m23">
<mml:mrow>
<mml:msup>
<mml:mi>c</mml:mi>
<mml:mo>&#x2032;</mml:mo>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula> and <inline-formula id="inf22">
<mml:math id="m24">
<mml:mrow>
<mml:msup>
<mml:mi>c</mml:mi>
<mml:mo>&#x2033;</mml:mo>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula> respectively. The coupling factor for subunit <inline-formula id="inf23">
<mml:math id="m25">
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> in channel <inline-formula id="inf24">
<mml:math id="m26">
<mml:mrow>
<mml:mi>c</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> is given by<disp-formula id="e3">
<mml:math id="m27">
<mml:mrow>
<mml:msub>
<mml:mi>&#x3b3;</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mi>exp</mml:mi>
<mml:mrow>
<mml:mfenced open="[" close="]" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3b4;</mml:mi>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:msubsup>
<mml:mi>s</mml:mi>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mi>c</mml:mi>
</mml:msubsup>
<mml:mo>&#x2b;</mml:mo>
<mml:msubsup>
<mml:mi>s</mml:mi>
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mo>&#x2b;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
<mml:mi>c</mml:mi>
</mml:msubsup>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x2b;</mml:mo>
<mml:msup>
<mml:mi>&#x3c3;</mml:mi>
<mml:mo>&#x2032;</mml:mo>
</mml:msup>
<mml:msubsup>
<mml:mi>s</mml:mi>
<mml:mi>j</mml:mi>
<mml:msup>
<mml:mi>c</mml:mi>
<mml:mo>&#x2032;</mml:mo>
</mml:msup>
</mml:msubsup>
<mml:mo>&#x2b;</mml:mo>
<mml:msup>
<mml:mi>&#x3c3;</mml:mi>
<mml:mo>&#x2033;</mml:mo>
</mml:msup>
<mml:msubsup>
<mml:mi>s</mml:mi>
<mml:mi>k</mml:mi>
<mml:msup>
<mml:mi>c</mml:mi>
<mml:mo>&#x2033;</mml:mo>
</mml:msup>
</mml:msubsup>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>,</mml:mo>
</mml:mrow>
</mml:math>
<label>(3)</label>
</disp-formula>where <inline-formula id="inf25">
<mml:math id="m28">
<mml:mrow>
<mml:msup>
<mml:mi>&#x3c3;</mml:mi>
<mml:mo>&#x2032;</mml:mo>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula> and <inline-formula id="inf26">
<mml:math id="m29">
<mml:mrow>
<mml:msup>
<mml:mi>&#x3c3;</mml:mi>
<mml:mo>&#x2033;</mml:mo>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula> represents the subunit interaction strength between subunit <inline-formula id="inf27">
<mml:math id="m30">
<mml:mrow>
<mml:mi>i</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> in channel <inline-formula id="inf28">
<mml:math id="m31">
<mml:mrow>
<mml:mi>c</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> and subunits <inline-formula id="inf29">
<mml:math id="m32">
<mml:mrow>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> and <inline-formula id="inf30">
<mml:math id="m33">
<mml:mrow>
<mml:mi>k</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> in neighboring channels <inline-formula id="inf31">
<mml:math id="m34">
<mml:mrow>
<mml:msup>
<mml:mi>c</mml:mi>
<mml:mo>&#x2032;</mml:mo>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula> and <inline-formula id="inf32">
<mml:math id="m35">
<mml:mrow>
<mml:msup>
<mml:mi>c</mml:mi>
<mml:mo>&#x2033;</mml:mo>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>. For simplicity, we will assume <inline-formula id="inf33">
<mml:math id="m36">
<mml:mrow>
<mml:msup>
<mml:mi>&#x3c3;</mml:mi>
<mml:mo>&#x2032;</mml:mo>
</mml:msup>
<mml:mo>&#x3d;</mml:mo>
<mml:msup>
<mml:mi>&#x3c3;</mml:mi>
<mml:mo>&#x2033;</mml:mo>
</mml:msup>
<mml:mo>&#x3d;</mml:mo>
<mml:mi>&#x3c3;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> in this study. However, it is likely that in realistic scenarios these parameters may differ due to variations in channel properties. With this formulation the transition rates have the same form as in <xref ref-type="disp-formula" rid="e1">Equation 1</xref> and remain consistent with the principle of detailed balance.</p>
<p>To model inter-RyR2 interactions within a cluster, we arrange RyR2 channels either with the oblique or adjoining configurations shown in <xref ref-type="fig" rid="F1">Figure 1</xref>. Each channel occupies a grid position labeled by coordinates <inline-formula id="inf34">
<mml:math id="m37">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:msub>
<mml:mi>n</mml:mi>
<mml:mi>x</mml:mi>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>n</mml:mi>
<mml:mi>y</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula>, and consists of four subunits labeled <inline-formula id="inf35">
<mml:math id="m38">
<mml:mrow>
<mml:msub>
<mml:mi>s</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>. Each subunit is uniquely identified by the triplet <inline-formula id="inf36">
<mml:math id="m39">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:msub>
<mml:mi>n</mml:mi>
<mml:mi>x</mml:mi>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>n</mml:mi>
<mml:mi>y</mml:mi>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>s</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula>, which denotes the lattice position and the state of the tetramer. The model incorporates two distinct types of coupling interactions. First, intra-channel coupling occurs between adjacent subunits within the same RyR2 tetramer, as described in <xref ref-type="disp-formula" rid="e2">Equation 2</xref>. Second, inter-channel coupling links specific subunits from neighbouring channels according to <xref ref-type="disp-formula" rid="e3">Equation 3</xref>. In the adjoining arrangement, subunits form connections with two nearest-neighbour channels, reflecting the side-by-side contact geometry. In the oblique arrangement, each subunit couples to only one nearest neighbour, corresponding to the staggered interaction pattern. To capture more realistic cluster arrangements, we will also consider disordered configurations that reflect the structural heterogeneity observed in cardiac myocytes. This computational framework will allow us to systematically investigate how the loss of structural integrity affects Ca release dynamics.</p>
</sec>
<sec id="s2-3">
<title>Model of Ca regulation of RyR2 cluster dynamics</title>
<p>Super-resolution imaging studies reveal that RyR2 cluster sizes span a wide range. The most recent work by <xref ref-type="bibr" rid="B17">Hou et al. (2015)</xref> reports many small clusters containing roughly 20&#x2013;50 channels (spanning approximately 150&#x2013;250 nm in diameter), while earlier studies by <xref ref-type="bibr" rid="B12">Galice et al. (2018)</xref> observed larger aggregates that could extend to several hundred RyR2s, with typical averages closer to 50&#x2013;100 channels (spanning approximately 250&#x2013;350 nm in diameter). Although the absolute estimates vary across techniques, the consistent finding is that ventricular myocytes contain a heterogeneous population of clusters, ranging from solitary channels and small groups to large assemblies. The functional behaviour of these clusters is strongly shaped by Ca diffusion in the narrow dyadic cleft. Ca diffuses rapidly in the cytosol, with an effective diffusion coefficient in the range <inline-formula id="inf37">
<mml:math id="m40">
<mml:mrow>
<mml:mi>D</mml:mi>
<mml:mo>&#x2248;</mml:mo>
<mml:mn>100</mml:mn>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>500</mml:mn>
<mml:msup>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3bc;</mml:mi>
<mml:mi>m</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mn>2</mml:mn>
</mml:msup>
<mml:mo>/</mml:mo>
<mml:mi>s</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> (<xref ref-type="bibr" rid="B6">Bers and Peskoff, 1991</xref>; <xref ref-type="bibr" rid="B34">Swietach et al., 2010</xref>). This wide range is due to the presence of multiple buffers and the structural complexity of the intracellular space. Given an RyR2 open time of roughly <inline-formula id="inf38">
<mml:math id="m41">
<mml:mrow>
<mml:mi>&#x3c4;</mml:mi>
<mml:mo>&#x223c;</mml:mo>
<mml:mn>1</mml:mn>
<mml:mi>m</mml:mi>
<mml:mi>s</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>, the expected diffusion length (<inline-formula id="inf39">
<mml:math id="m42">
<mml:mrow>
<mml:mi>d</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:msqrt>
<mml:mrow>
<mml:mn>4</mml:mn>
<mml:mi>D</mml:mi>
<mml:mi>&#x3c4;</mml:mi>
</mml:mrow>
</mml:msqrt>
</mml:mrow>
</mml:math>
</inline-formula>) is in the range <inline-formula id="inf40">
<mml:math id="m43">
<mml:mrow>
<mml:mn>600</mml:mn>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>1000</mml:mn>
<mml:mi>n</mml:mi>
<mml:mi>m</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>. This length scale exceeds the size of even the largest RyR2 clusters <inline-formula id="inf41">
<mml:math id="m44">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mo>&#x2264;</mml:mo>
<mml:mn>400</mml:mn>
<mml:mi>n</mml:mi>
<mml:mi>m</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula>, implying that Ca released from a single open channel rapidly equilibrates across the entire cluster. As a result, RyR2s within a cluster are functionally coupled through the shared local Ca signal, independently of direct molecular contact.</p>
<p>Since diffusion is much faster than channel gating, we invoke the rapid diffusion approximation, which assumes Ca is uniform within the dyad. Under this approximation, the local Ca concentration is given by<disp-formula id="e4">
<mml:math id="m45">
<mml:mrow>
<mml:mrow>
<mml:mfenced open="[" close="]" separators="&#x7c;">
<mml:mrow>
<mml:mi>C</mml:mi>
<mml:mi>a</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x3d;</mml:mo>
<mml:msub>
<mml:mi>c</mml:mi>
<mml:mi>o</mml:mi>
</mml:msub>
<mml:mo>&#x2b;</mml:mo>
<mml:mi>g</mml:mi>
<mml:mo>&#xb7;</mml:mo>
<mml:msub>
<mml:mi>n</mml:mi>
<mml:mi>o</mml:mi>
</mml:msub>
<mml:mo>,</mml:mo>
</mml:mrow>
</mml:math>
<label>(4)</label>
</disp-formula>where <inline-formula id="inf42">
<mml:math id="m46">
<mml:mrow>
<mml:msub>
<mml:mi>c</mml:mi>
<mml:mn>0</mml:mn>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is the diastolic Ca concentration in the bulk cytosol, <inline-formula id="inf43">
<mml:math id="m47">
<mml:mrow>
<mml:msub>
<mml:mi>n</mml:mi>
<mml:mi>o</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> represents the number of open RyR2 channels within the cluster, and <inline-formula id="inf44">
<mml:math id="m48">
<mml:mrow>
<mml:mi>g</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> represents the Ca concentration increase in the dyadic cleft due to a single open RyR2 channel (<xref ref-type="bibr" rid="B6">Bers and Peskoff, 1991</xref>; <xref ref-type="bibr" rid="B22">Langer and Peskoff, 1996</xref>; <xref ref-type="bibr" rid="B2">Asfaw et al., 2013</xref>). Several computational studies of CICR have argued that [Ca] must reach relatively high concentrations in the dyadic junction. <xref ref-type="bibr" rid="B8">Cannell et al. (2013)</xref> argued that minimum dyadic [Ca] of <inline-formula id="inf45">
<mml:math id="m49">
<mml:mrow>
<mml:mo>&#x223c;</mml:mo>
<mml:mn>50</mml:mn>
<mml:mi>&#x3bc;</mml:mi>
<mml:mi>M</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> was required to maintain CICR in rat ventricular myocytes, while <xref ref-type="bibr" rid="B32">Shen et al. (2022)</xref> used half-maximal RyR2 activation concentrations (<inline-formula id="inf46">
<mml:math id="m50">
<mml:mrow>
<mml:msub>
<mml:mi>K</mml:mi>
<mml:mi>d</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>) ranging from <inline-formula id="inf47">
<mml:math id="m51">
<mml:mrow>
<mml:mn>25</mml:mn>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>45</mml:mn>
<mml:mi>&#x3bc;</mml:mi>
<mml:mi>M</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> in their models of RyR cluster dispersion during &#x3b2;-adrenergic stimulation. In this study we will set <inline-formula id="inf48">
<mml:math id="m52">
<mml:mrow>
<mml:mi>g</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>25</mml:mn>
<mml:mi>&#x3bc;</mml:mi>
<mml:mi>M</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>, so that 1-3 open channels produce dyadic [Ca] in the range of 25&#x2013;75 &#x3bc;M, consistent with the concentrations required to support CICR in these studies.</p>
<p>To determine the transition rates in our model, we assume first-order kinetics with respect to cytosolic Ca. Specifically, the forward rate for subunit opening is given by <inline-formula id="inf49">
<mml:math id="m53">
<mml:mrow>
<mml:msub>
<mml:mi>k</mml:mi>
<mml:mi>f</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:msubsup>
<mml:mi>k</mml:mi>
<mml:mi>f</mml:mi>
<mml:mi>o</mml:mi>
</mml:msubsup>
<mml:mrow>
<mml:mfenced open="[" close="]" separators="&#x7c;">
<mml:mrow>
<mml:mi>C</mml:mi>
<mml:mi>a</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:mrow>
</mml:math>
</inline-formula>, where <inline-formula id="inf50">
<mml:math id="m54">
<mml:mrow>
<mml:msubsup>
<mml:mi>k</mml:mi>
<mml:mi>f</mml:mi>
<mml:mi>o</mml:mi>
</mml:msubsup>
</mml:mrow>
</mml:math>
</inline-formula> is a binding rate constant and <inline-formula id="inf51">
<mml:math id="m55">
<mml:mrow>
<mml:mfenced open="[" close="]" separators="&#x7c;">
<mml:mrow>
<mml:mi>C</mml:mi>
<mml:mi>a</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> is the local Ca concentration. We set <inline-formula id="inf52">
<mml:math id="m56">
<mml:mrow>
<mml:msubsup>
<mml:mi>k</mml:mi>
<mml:mi>f</mml:mi>
<mml:mi>o</mml:mi>
</mml:msubsup>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>0.005</mml:mn>
<mml:mtext>&#x2009;</mml:mtext>
<mml:msup>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3bc;</mml:mi>
<mml:mi>M</mml:mi>
<mml:mo>&#xb7;</mml:mo>
<mml:mi>m</mml:mi>
<mml:mi>s</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mrow>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula> to ensure that RyR2 channels remain predominantly closed at a diastolic concentratonof <inline-formula id="inf53">
<mml:math id="m57">
<mml:mrow>
<mml:mrow>
<mml:mfenced open="[" close="]" separators="&#x7c;">
<mml:mrow>
<mml:mi>C</mml:mi>
<mml:mi>a</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mo>&#x223c;</mml:mo>
<mml:mn>0.1</mml:mn>
<mml:mi>&#x3bc;</mml:mi>
<mml:mi>M</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>. This choice is consistent with experimental observations, see for example <xref ref-type="bibr" rid="B32">Shen et al. (2022)</xref>, showing that spontaneous Ca sparks are extremely rare at baseline (<inline-formula id="inf54">
<mml:math id="m58">
<mml:mrow>
<mml:mo>&#x223c;</mml:mo>
<mml:mn>0.1</mml:mn>
<mml:mi>s</mml:mi>
<mml:mi>p</mml:mi>
<mml:mi>a</mml:mi>
<mml:mi>r</mml:mi>
<mml:mi>k</mml:mi>
<mml:mi>s</mml:mi>
<mml:mo>/</mml:mo>
<mml:mn>100</mml:mn>
<mml:mi>&#x3bc;</mml:mi>
<mml:mi>m</mml:mi>
<mml:mo>/</mml:mo>
<mml:mi>s</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>). Although this forward rate is substantially lower than binding rates measured in bilayer recordings (<xref ref-type="bibr" rid="B28">Mukherjee et al., 2012</xref>; <xref ref-type="bibr" rid="B39">Xu and Meissner, 2004</xref>), the difference reflects the highly regulated environment of intact cells. In particular, physiological <inline-formula id="inf55">
<mml:math id="m59">
<mml:mrow>
<mml:mi>M</mml:mi>
<mml:msup>
<mml:mi>g</mml:mi>
<mml:mrow>
<mml:mn>2</mml:mn>
<mml:mo>&#x2b;</mml:mo>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula> occupies the Ca-activation sites at resting [Ca] and must be displaced by Ca to allow channel opening, effectively reducing the Ca sensitivity and stabilizing the closed state compared to isolated channels in bilayers (<xref ref-type="bibr" rid="B23">Laver et al., 1997</xref>; <xref ref-type="bibr" rid="B36">Walweel and Laver, 2015</xref>; <xref ref-type="bibr" rid="B40">Zahradn&#xed;kov&#xe1; et al., 2025</xref>). The backward rate for subunit closing is Ca-independent and fixed at <inline-formula id="inf56">
<mml:math id="m60">
<mml:mrow>
<mml:msub>
<mml:mi>k</mml:mi>
<mml:mi>b</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>0.3</mml:mn>
<mml:mtext>&#x2009;</mml:mtext>
<mml:msup>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>m</mml:mi>
<mml:mi>s</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mrow>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula> (<xref ref-type="bibr" rid="B39">Xu and Meissner, 2004</xref>), which corresponds to an average open time of approximately <inline-formula id="inf57">
<mml:math id="m61">
<mml:mrow>
<mml:mn>3</mml:mn>
<mml:mi>m</mml:mi>
<mml:mi>s</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>. Subunit cooperativity is captured by a dimensionless mismatch penalty <inline-formula id="inf58">
<mml:math id="m62">
<mml:mrow>
<mml:mi>&#x3b4;</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>0.5</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>, which stabilizes the closed conformation at rest without imposing such a large energetic cost that the channel becomes unresponsive at elevated Ca. A pore conducts only when at least three of the four subunits are open, a majority rule that preserves observed closed-time statistics and prevents spurious openings (<xref ref-type="bibr" rid="B15">Greene et al., 2023</xref>). Collectively, these parameters produce physiologically plausible single-channel kinetics and cluster-level behavior across the physiological Ca range.</p>
</sec>
<sec id="s2-4">
<title>Stochastic simulation algorithm</title>
<p>The Gillespie algorithm is implemented as a stochastic simulation method to model the exact temporal evolution of RyR2 channel gating within clusters (<xref ref-type="bibr" rid="B14">Gillespie, 2007</xref>). At each time step, the algorithm calculates the total reaction rate across all subunits in the system based on their current states and local coupling environments. The time to the next reaction is drawn from an exponential distribution with parameter equal to the total rate, ensuring proper stochastic timing. A specific reaction is then selected using weighted random selection proportional to individual subunit rates, after which the chosen subunit transitions between closed and open states. This process continues iteratively, with the system state analyzed after each reaction to count the number of open channels.</p>
</sec>
</sec>
<sec sec-type="results" id="s3">
<title>Results</title>
<sec id="s3-1">
<title>Measuring the timing of a spontaneous Ca spark</title>
<p>To study the stochastic dynamics of RyR2 clusters, we first simulated the time evolution of a <inline-formula id="inf59">
<mml:math id="m63">
<mml:mrow>
<mml:mn>10</mml:mn>
<mml:mo>&#xd7;</mml:mo>
<mml:mn>10</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula> array of RyR2 channels using our stochastic simulation algorithm. To monitor the cluster, we keep track of the number of open RyR2 channels, denoted as <inline-formula id="inf60">
<mml:math id="m64">
<mml:mrow>
<mml:msub>
<mml:mi>n</mml:mi>
<mml:mi>o</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>, which is defined as the number of channels where 3 or more subunits are in the open state. <xref ref-type="fig" rid="F2">Figure 2</xref> shows <inline-formula id="inf61">
<mml:math id="m65">
<mml:mrow>
<mml:msub>
<mml:mi>n</mml:mi>
<mml:mi>o</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> as function of time for three simulation runs where the cytosolic Ca concentration is fixed at <inline-formula id="inf62">
<mml:math id="m66">
<mml:mrow>
<mml:msub>
<mml:mi>c</mml:mi>
<mml:mi>o</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>6.0</mml:mn>
<mml:mi>&#x3bc;</mml:mi>
<mml:mi>M</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>. Each independent simulation run is shown as a red, black, and blue line. In these runs the cluster stays mostly closed and occasionally one or two channels transition to the open state. After a waiting period a large fluctuation pushes <inline-formula id="inf63">
<mml:math id="m67">
<mml:mrow>
<mml:msub>
<mml:mi>n</mml:mi>
<mml:mi>o</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> above a critical threshold, leading to a rapid cooperative avtivation of the entire cluster. Once this transition occurs, the number of open channels grows explosively from a few <inline-formula id="inf64">
<mml:math id="m68">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:msub>
<mml:mi>n</mml:mi>
<mml:mi>o</mml:mi>
</mml:msub>
<mml:mo>&#x223c;</mml:mo>
<mml:mn>0</mml:mn>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> to nearly all the channels in the cluster <inline-formula id="inf65">
<mml:math id="m69">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:msub>
<mml:mi>n</mml:mi>
<mml:mi>o</mml:mi>
</mml:msub>
<mml:mo>&#x223c;</mml:mo>
<mml:mn>100</mml:mn>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula>. This behaviour arises because the local Ca concentration is dependent on <inline-formula id="inf66">
<mml:math id="m70">
<mml:mrow>
<mml:msub>
<mml:mi>n</mml:mi>
<mml:mi>o</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> according to <xref ref-type="disp-formula" rid="e4">Equation 4</xref>. Thus, the opening rate of channels increases in a strongly nonlinear fashion due to subunit cooperativity within and between RyR2 channels.</p>
<fig id="F2" position="float">
<label>FIGURE 2</label>
<caption>
<p>Stochastic activation of a <inline-formula id="inf67">
<mml:math id="m71">
<mml:mrow>
<mml:mn>10</mml:mn>
<mml:mo>&#xd7;</mml:mo>
<mml:mn>10</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula> RyR2 cluster connected in the oblique configuration. The cytosolic Ca concentration is fixed at <inline-formula id="inf68">
<mml:math id="m72">
<mml:mrow>
<mml:msub>
<mml:mi>c</mml:mi>
<mml:mi>o</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>6.0</mml:mn>
<mml:mi>&#x3bc;</mml:mi>
<mml:mi>M</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>. Three independent simulation runs are shown corresponding to the red, black, and blue lines. The activation time <inline-formula id="inf69">
<mml:math id="m73">
<mml:mrow>
<mml:mi>T</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> (for the simulation shown in black) is the time when the channels are all closed, to when <inline-formula id="inf70">
<mml:math id="m74">
<mml:mrow>
<mml:msub>
<mml:mi>n</mml:mi>
<mml:mi>o</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> exceeds <inline-formula id="inf71">
<mml:math id="m75">
<mml:mrow>
<mml:msub>
<mml:mi>n</mml:mi>
<mml:mi>c</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>5</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>. The parameters used in this simulation are <inline-formula id="inf72">
<mml:math id="m76">
<mml:mrow>
<mml:mi>&#x3b4;</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>0.5</mml:mn>
<mml:mo>,</mml:mo>
<mml:mi>&#x3c3;</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>0.5</mml:mn>
<mml:mo>,</mml:mo>
<mml:mi>g</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>25</mml:mn>
<mml:mi>&#x3bc;</mml:mi>
<mml:mi>M</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>,<inline-formula id="inf208">
<mml:math id="m212">
<mml:mrow>
<mml:msubsup>
<mml:mi>k</mml:mi>
<mml:mi>f</mml:mi>
<mml:mi>o</mml:mi>
</mml:msubsup>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>0.005</mml:mn>
<mml:mtext>&#x2009;</mml:mtext>
<mml:msup>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3bc;</mml:mi>
<mml:mi>M</mml:mi>
<mml:mo>&#xb7;</mml:mo>
<mml:mi>m</mml:mi>
<mml:mi>s</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mrow>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula>, and <inline-formula id="inf73">
<mml:math id="m77">
<mml:mrow>
<mml:msub>
<mml:mi>k</mml:mi>
<mml:mi>b</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>0.3</mml:mn>
<mml:mtext>&#x2009;</mml:mtext>
<mml:msup>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>m</mml:mi>
<mml:mi>s</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mrow>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:msup>
<mml:mo>.</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula>
</p>
</caption>
<graphic xlink:href="fphys-16-1731863-g002.tif">
<alt-text content-type="machine-generated">Bar graph displaying stimulus occurrences over time in milliseconds. Red, blue, and black bars represent different periods, with annotations \( n_o &#x3d; 3 \), and \( T \). A dashed red line at \( n_o &#x3d; 5 \) emphasizes a reference level.</alt-text>
</graphic>
</fig>
<p>The rapid increase in <inline-formula id="inf74">
<mml:math id="m78">
<mml:mrow>
<mml:msub>
<mml:mi>n</mml:mi>
<mml:mi>o</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> corresponds to the initiation of a Ca spark. Since the excitation occurred because of stochastic fluctuations, we will refer to these events as spontaneous Ca sparks. After activation, RyR2 channels proceed to inactivate and return to the closed state. However, this inactivation process occurs on a much shorter timescale than the waiting time to a spontaneous Ca spark and is not modelled in this study. To quantify the spontaneous Ca spark initiation process, we define the activation time <inline-formula id="inf75">
<mml:math id="m79">
<mml:mrow>
<mml:mi>T</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>, shown in <xref ref-type="fig" rid="F2">Figure 2</xref> for the simulation run corresponding to the black line, as the time required for the cluster to undergo the initial transition from the closed state to a fully activated cluster. Specifically, <inline-formula id="inf76">
<mml:math id="m80">
<mml:mrow>
<mml:mi>T</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> is measured from <inline-formula id="inf77">
<mml:math id="m81">
<mml:mrow>
<mml:mi>t</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>0</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>, when all RyR2 channels in the cluster are initially closed (all subunits are in the state <inline-formula id="inf78">
<mml:math id="m82">
<mml:mrow>
<mml:msub>
<mml:mi>s</mml:mi>
<mml:mi>i</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>), to the moment when <inline-formula id="inf79">
<mml:math id="m83">
<mml:mrow>
<mml:msub>
<mml:mi>n</mml:mi>
<mml:mi>o</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> exceeds a critical threshold <inline-formula id="inf80">
<mml:math id="m84">
<mml:mrow>
<mml:msub>
<mml:mi>n</mml:mi>
<mml:mi>c</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>. In general, <inline-formula id="inf81">
<mml:math id="m85">
<mml:mrow>
<mml:msub>
<mml:mi>n</mml:mi>
<mml:mi>c</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> should be picked to be larger than a typical fluctuation, and in this study we will use <inline-formula id="inf82">
<mml:math id="m86">
<mml:mrow>
<mml:msub>
<mml:mi>n</mml:mi>
<mml:mi>c</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>5</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>. However, we note here that our results are independent of <inline-formula id="inf83">
<mml:math id="m87">
<mml:mrow>
<mml:msub>
<mml:mi>n</mml:mi>
<mml:mi>c</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> providing <inline-formula id="inf84">
<mml:math id="m88">
<mml:mrow>
<mml:msub>
<mml:mi>n</mml:mi>
<mml:mi>c</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is larger than the baseline fluctuations. For example, in the simulation shown in <xref ref-type="fig" rid="F2">Figure 2</xref>, any <inline-formula id="inf85">
<mml:math id="m89">
<mml:mrow>
<mml:msub>
<mml:mi>n</mml:mi>
<mml:mi>c</mml:mi>
</mml:msub>
<mml:mo>&#x2265;</mml:mo>
<mml:mn>5</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula> gives the same waiting time since once <inline-formula id="inf86">
<mml:math id="m90">
<mml:mrow>
<mml:msub>
<mml:mi>n</mml:mi>
<mml:mi>c</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>5</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula> is reached then excitation proceeds with high likelihood. This approach provides a consistent measure of the waiting time before spontaneous Ca spark initiation, which determines the average frequency of spontaneous Ca sparks in the cell.</p>
</sec>
<sec id="s3-2">
<title>Control of spontaneous spark timing by diastolic Ca and cluster coupling</title>
<p>We next investigated how the mean waiting time <inline-formula id="inf87">
<mml:math id="m91">
<mml:mrow>
<mml:mi>T</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> to a spontaneous Ca spark depends on the dyadic Ca concentration <inline-formula id="inf88">
<mml:math id="m92">
<mml:mrow>
<mml:msub>
<mml:mi>c</mml:mi>
<mml:mi>o</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>. This concentration is typically <inline-formula id="inf89">
<mml:math id="m93">
<mml:mrow>
<mml:msub>
<mml:mi>c</mml:mi>
<mml:mi>o</mml:mi>
</mml:msub>
<mml:mo>&#x223c;</mml:mo>
<mml:mn>0.1</mml:mn>
<mml:mi>&#x3bc;</mml:mi>
<mml:mi>M</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> during rest, but will increase substantially when there is a local opening of a nearby LCC channel. <xref ref-type="fig" rid="F3">Figure 3A</xref> shows the relationship between <inline-formula id="inf90">
<mml:math id="m94">
<mml:mrow>
<mml:mi>T</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> and <inline-formula id="inf91">
<mml:math id="m95">
<mml:mrow>
<mml:msub>
<mml:mi>c</mml:mi>
<mml:mi>o</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> for the oblique configuration of the cluster. Here, we vary <inline-formula id="inf92">
<mml:math id="m96">
<mml:mrow>
<mml:msub>
<mml:mi>c</mml:mi>
<mml:mi>o</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> from <inline-formula id="inf93">
<mml:math id="m97">
<mml:mrow>
<mml:mn>5.0</mml:mn>
<mml:mi>&#x3bc;</mml:mi>
<mml:mi>M</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> to <inline-formula id="inf94">
<mml:math id="m98">
<mml:mrow>
<mml:mn>120</mml:mn>
<mml:mi>&#x3bc;</mml:mi>
<mml:mi>M</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>, which represents the typical Ca concentrations at a dyadic junction that occurs during LCC activation. Three values of the inter-subunit coupling parameter <inline-formula id="inf95">
<mml:math id="m99">
<mml:mrow>
<mml:mi>&#x3c3;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> are examined: <inline-formula id="inf96">
<mml:math id="m100">
<mml:mrow>
<mml:mi>&#x3c3;</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>0</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula> (no coupling), <inline-formula id="inf97">
<mml:math id="m101">
<mml:mrow>
<mml:mi>&#x3c3;</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>0.5</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula> (moderate coupling), and <inline-formula id="inf98">
<mml:math id="m102">
<mml:mrow>
<mml:mi>&#x3c3;</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula> (strong coupling). The results show that <inline-formula id="inf99">
<mml:math id="m103">
<mml:mrow>
<mml:mi>T</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> changes exponentially with the concentration <inline-formula id="inf100">
<mml:math id="m104">
<mml:mrow>
<mml:msub>
<mml:mi>c</mml:mi>
<mml:mi>o</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>. At high diastolic Ca (<inline-formula id="inf101">
<mml:math id="m105">
<mml:mrow>
<mml:msub>
<mml:mi>c</mml:mi>
<mml:mi>o</mml:mi>
</mml:msub>
<mml:mo>&#x223c;</mml:mo>
<mml:mn>100</mml:mn>
<mml:mi>&#x3bc;</mml:mi>
<mml:mi>M</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>) all three cases have comparable firing times in the range <inline-formula id="inf102">
<mml:math id="m106">
<mml:mrow>
<mml:mn>0.5</mml:mn>
<mml:mi>m</mml:mi>
<mml:mi>s</mml:mi>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>2</mml:mn>
<mml:mi>m</mml:mi>
<mml:mi>s</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>. In contrast, at lower Ca concentrations the waiting time increases exponentially with decreasing <inline-formula id="inf103">
<mml:math id="m107">
<mml:mrow>
<mml:msub>
<mml:mi>c</mml:mi>
<mml:mi>o</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>. In particular, when subunit interactions between channels is engaged, with <inline-formula id="inf104">
<mml:math id="m108">
<mml:mrow>
<mml:mi>&#x3c3;</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1.0</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>, the waiting time increases from <inline-formula id="inf105">
<mml:math id="m109">
<mml:mrow>
<mml:mo>&#x223c;</mml:mo>
<mml:mn>2</mml:mn>
<mml:mi>m</mml:mi>
<mml:mi>s</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> at <inline-formula id="inf106">
<mml:math id="m110">
<mml:mrow>
<mml:msub>
<mml:mi>c</mml:mi>
<mml:mi>o</mml:mi>
</mml:msub>
<mml:mo>&#x223c;</mml:mo>
<mml:mn>120</mml:mn>
<mml:mi>&#x3bc;</mml:mi>
<mml:mi>M</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>, to <inline-formula id="inf107">
<mml:math id="m111">
<mml:mrow>
<mml:mo>&#x223c;</mml:mo>
<mml:msup>
<mml:mn>10</mml:mn>
<mml:mn>5</mml:mn>
</mml:msup>
<mml:mi>m</mml:mi>
<mml:mi>s</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> at <inline-formula id="inf108">
<mml:math id="m112">
<mml:mrow>
<mml:msub>
<mml:mi>c</mml:mi>
<mml:mi>o</mml:mi>
</mml:msub>
<mml:mo>&#x223c;</mml:mo>
<mml:mn>5.0</mml:mn>
<mml:mi>&#x3bc;</mml:mi>
<mml:mi>M</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>. <xref ref-type="fig" rid="F3">Figure 3B</xref> shows analogous results for the adjoining configuration. The qualitative features are the same as for the oblique case, but the dependence on <inline-formula id="inf109">
<mml:math id="m113">
<mml:mrow>
<mml:mi>&#x3c3;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> is more pronounced, since each subunit has two contacts rather than just one. In <xref ref-type="fig" rid="F3">Figures 3C,D</xref> we compute <inline-formula id="inf110">
<mml:math id="m114">
<mml:mrow>
<mml:mi>T</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>, for both the oblique and adjoining configurations respectively, for cluster sizes of <inline-formula id="inf111">
<mml:math id="m115">
<mml:mrow>
<mml:mn>5</mml:mn>
<mml:mo>&#xd7;</mml:mo>
<mml:mn>5</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>, <inline-formula id="inf112">
<mml:math id="m116">
<mml:mrow>
<mml:mn>10</mml:mn>
<mml:mo>&#xd7;</mml:mo>
<mml:mn>10</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>, and <inline-formula id="inf113">
<mml:math id="m117">
<mml:mrow>
<mml:mn>20</mml:mn>
<mml:mo>&#xd7;</mml:mo>
<mml:mn>20</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>. These results shows that the dependence of <inline-formula id="inf114">
<mml:math id="m118">
<mml:mrow>
<mml:mi>T</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> on <inline-formula id="inf115">
<mml:math id="m119">
<mml:mrow>
<mml:msub>
<mml:mi>c</mml:mi>
<mml:mi>o</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> is similar across the wide range of cluster sizes that are expected to be found in cardiac myocytes.</p>
<fig id="F3" position="float">
<label>FIGURE 3</label>
<caption>
<p>Dependence of spontaneous spark timing on diastolic Ca <inline-formula id="inf116">
<mml:math id="m120">
<mml:mrow>
<mml:msub>
<mml:mi>c</mml:mi>
<mml:mi>o</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula>. <bold>(A)</bold> Mean waiting time <inline-formula id="inf117">
<mml:math id="m121">
<mml:mrow>
<mml:mi>T</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> to a spontaneous Ca spark as a function of diastolic Ca concentration <inline-formula id="inf118">
<mml:math id="m122">
<mml:mrow>
<mml:msub>
<mml:mi>c</mml:mi>
<mml:mi>o</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> for the oblique cluster configuration. <inline-formula id="inf119">
<mml:math id="m123">
<mml:mrow>
<mml:mi>T</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> is computed by averaging over 50 independent simulation runs. Error bars shown are computed as the standard deviation of 5 independent sets of 10 simulation runs each. <bold>(B)</bold> Same simulation as in A but with RyR2 channels arranged in the adjoining configuration. <bold>(C)</bold> Same simulations as in A but with cluster sizes of <inline-formula id="inf120">
<mml:math id="m124">
<mml:mrow>
<mml:mn>5</mml:mn>
<mml:mo>&#xd7;</mml:mo>
<mml:mn>5</mml:mn>
<mml:mo>,</mml:mo>
<mml:mn>10</mml:mn>
<mml:mo>&#xd7;</mml:mo>
<mml:mn>10</mml:mn>
<mml:mo>,</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> and <inline-formula id="inf121">
<mml:math id="m125">
<mml:mrow>
<mml:mn>20</mml:mn>
<mml:mo>&#xd7;</mml:mo>
<mml:mn>20</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>. <bold>(D)</bold> Same simulations as in B with the indicated cluster sizes. For all simulations in C and D we have fixed <inline-formula id="inf122">
<mml:math id="m126">
<mml:mrow>
<mml:mi>&#x3c3;</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>0.5</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>.</p>
</caption>
<graphic xlink:href="fphys-16-1731863-g003.tif">
<alt-text content-type="machine-generated">Graphs labeled A to D show the relationship between T (in milliseconds) and c&#x2080; (in micromolar). Graphs A and B use colors blue, red, and black for &#x3C3; values of one, 0.5, and zero, respectively. Graphs C and D use purple, green, and orange for dimensions 20x20, 10x10, and 5x5. All graphs display a decreasing trend.</alt-text>
</graphic>
</fig>
<p>To further explore the role of coupling, we examined the dependence of <inline-formula id="inf123">
<mml:math id="m127">
<mml:mrow>
<mml:mi>T</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> on <inline-formula id="inf124">
<mml:math id="m128">
<mml:mrow>
<mml:mi>&#x3c3;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> at two fixed Ca concentrations. <xref ref-type="fig" rid="F4">Figure 4A</xref> shows results at a resting concentration of <inline-formula id="inf125">
<mml:math id="m129">
<mml:mrow>
<mml:msub>
<mml:mi>c</mml:mi>
<mml:mi>o</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>5.0</mml:mn>
<mml:mi>&#x3bc;</mml:mi>
<mml:mi>M</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>. In this regime, the mean waiting time measured for both the oblique and adjoining configurations grows almost three orders of magnitude as <inline-formula id="inf126">
<mml:math id="m130">
<mml:mrow>
<mml:mi>&#x3c3;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> is increased. In contrast, at a diastolic Ca concentration of <inline-formula id="inf127">
<mml:math id="m131">
<mml:mrow>
<mml:msub>
<mml:mi>c</mml:mi>
<mml:mi>o</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>100</mml:mn>
<mml:mi>&#x3bc;</mml:mi>
<mml:mi>M</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>, shown in <xref ref-type="fig" rid="F4">Figure 4B</xref>, <inline-formula id="inf128">
<mml:math id="m132">
<mml:mrow>
<mml:mi>T</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> increases only modestly over a few milliseconds. Together, these findings identify inter-channel coupling as a powerful mechanism for tuning cluster stability at low Ca concentrations. At high Ca, clusters activate rapidly and reliably, with little dependence on coupling strength. At low Ca, however, even modest increases in <inline-formula id="inf129">
<mml:math id="m133">
<mml:mrow>
<mml:mi>&#x3c3;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> dramatically stabilize the closed state by exponentially prolonging the waiting time <inline-formula id="inf130">
<mml:math id="m134">
<mml:mrow>
<mml:mi>T</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>. This dual behaviour ensures that RyR2 clusters remain quiescent at rest, while still capable of fast activation when triggered during excitation&#x2013;contraction coupling.</p>
<fig id="F4" position="float">
<label>FIGURE 4</label>
<caption>
<p>Dependence of spontaneous Ca spark timing on inter-subunit coupling strength. <bold>(A)</bold> Dependence of <inline-formula id="inf131">
<mml:math id="m135">
<mml:mrow>
<mml:mi>T</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> on <inline-formula id="inf132">
<mml:math id="m136">
<mml:mrow>
<mml:mi>&#x3c3;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> at <inline-formula id="inf133">
<mml:math id="m137">
<mml:mrow>
<mml:msub>
<mml:mi>c</mml:mi>
<mml:mi>o</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>5.0</mml:mn>
<mml:mi>&#x3bc;</mml:mi>
<mml:mi>M</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> for a <inline-formula id="inf134">
<mml:math id="m138">
<mml:mrow>
<mml:mn>10</mml:mn>
<mml:mo>&#xd7;</mml:mo>
<mml:mn>10</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula> cluster. Red (black) line corresponds to the adjoining (oblique) configuration. <bold>(B)</bold> Dependence of <inline-formula id="inf135">
<mml:math id="m139">
<mml:mrow>
<mml:mi>T</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> on <inline-formula id="inf136">
<mml:math id="m140">
<mml:mrow>
<mml:mi>&#x3c3;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> at high Ca (<inline-formula id="inf137">
<mml:math id="m141">
<mml:mrow>
<mml:mfenced open="" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mrow>
<mml:msub>
<mml:mi>c</mml:mi>
<mml:mi>o</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>100</mml:mn>
<mml:mi>&#x3bc;</mml:mi>
<mml:mi>M</mml:mi>
</mml:mrow>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula>. The waiting time increases only modestly, indicating weak coupling effects at elevated Ca. Points plotted are averaged over 50 samples and the error bar error bars shown are estimated by computing the standard deviation of 5 independent sets of 10 runs each.</p>
</caption>
<graphic xlink:href="fphys-16-1731863-g004.tif">
<alt-text content-type="machine-generated">Two line graphs show the relationship between \( T \) in milliseconds and \( \sigma \). Graph A shows two lines: a red line with square points increasing sharply, and a black line with circular points increasing steadily. Graph B has a similar pattern but on a smaller scale, with both lines increasing, the red line more steeply. Both graphs include error bars.</alt-text>
</graphic>
</fig>
</sec>
<sec id="s3-3">
<title>Modeling heterogeneous clusters</title>
<p>Experimental studies have revealed that RyR2 clusters undergo fractionation and lose their structural integrity in diseased states. To investigate the functional consequences of cluster fractionation, we developed a computational model that simulates the progressive disruption of RyR2 cluster connectivity. Our approach implements stochastic bond breaking between adjacent RyR2 channels within the cluster. Specifically, we assign a probability <inline-formula id="inf138">
<mml:math id="m142">
<mml:mrow>
<mml:mi>p</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> that a connection between two nearest-neighbor subunits remains intact. The parameter <inline-formula id="inf139">
<mml:math id="m143">
<mml:mrow>
<mml:mi>p</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> serves as a measure of cluster integrity, where <inline-formula id="inf140">
<mml:math id="m144">
<mml:mrow>
<mml:mi>p</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula> corresponds to a fully connected cluster with all inter-channel connections preserved, while <inline-formula id="inf141">
<mml:math id="m145">
<mml:mrow>
<mml:mi>p</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>0</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula> represents a completely dissociated cluster with all connections broken. For each value of <inline-formula id="inf142">
<mml:math id="m146">
<mml:mrow>
<mml:mi>p</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>, we generate an ensemble of connectivity matrices that define the network topology and compute the average <inline-formula id="inf143">
<mml:math id="m147">
<mml:mrow>
<mml:mi>T</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> over the ensemble. In <xref ref-type="fig" rid="F5">Figure 5</xref>, we show the relationship between cluster connectivity and spontaneous activation by plotting <inline-formula id="inf144">
<mml:math id="m148">
<mml:mrow>
<mml:mi>T</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> versus the connection probability <inline-formula id="inf145">
<mml:math id="m149">
<mml:mrow>
<mml:mi>p</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> at a Ca concentration of <inline-formula id="inf146">
<mml:math id="m150">
<mml:mrow>
<mml:msub>
<mml:mi>c</mml:mi>
<mml:mi>o</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>2.0</mml:mn>
<mml:mi>&#x3bc;</mml:mi>
<mml:mi>M</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>. The results show that the waiting time <inline-formula id="inf147">
<mml:math id="m151">
<mml:mrow>
<mml:mi>T</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> grows exponentially with increasing connection probability <inline-formula id="inf148">
<mml:math id="m152">
<mml:mrow>
<mml:mi>p</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>. In particular, for the case of the adjoining configuration, we find that an uncoupled cluster of <inline-formula id="inf149">
<mml:math id="m153">
<mml:mrow>
<mml:mn>10</mml:mn>
<mml:mo>&#xd7;</mml:mo>
<mml:mn>10</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula> RyR2s has a mean firing time of <inline-formula id="inf150">
<mml:math id="m154">
<mml:mrow>
<mml:mi>T</mml:mi>
<mml:mo>&#x223c;</mml:mo>
<mml:msup>
<mml:mn>10</mml:mn>
<mml:mn>3</mml:mn>
</mml:msup>
<mml:mi>m</mml:mi>
<mml:mi>s</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>. However, as <inline-formula id="inf151">
<mml:math id="m155">
<mml:mrow>
<mml:mi>p</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> is increased to just <inline-formula id="inf152">
<mml:math id="m156">
<mml:mrow>
<mml:mi>p</mml:mi>
<mml:mo>&#x223c;</mml:mo>
<mml:mn>0.5</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>, the waiting time rises to <inline-formula id="inf153">
<mml:math id="m157">
<mml:mrow>
<mml:mi>T</mml:mi>
<mml:mo>&#x223c;</mml:mo>
<mml:msup>
<mml:mn>10</mml:mn>
<mml:mn>6</mml:mn>
</mml:msup>
<mml:mi>m</mml:mi>
<mml:mi>s</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>. This, result shows that the spontaneous Ca spark frequency is exponentially sensitive to the probability that neighboring RyR2s are connected.</p>
<fig id="F5" position="float">
<label>FIGURE 5</label>
<caption>
<p>Dependence of spontaneous spark timing on cluster connectivity. Mean waiting time <inline-formula id="inf154">
<mml:math id="m158">
<mml:mrow>
<mml:mi>T</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> to a spontaneous Ca spark as function of the connection probability <inline-formula id="inf155">
<mml:math id="m159">
<mml:mrow>
<mml:mi>p</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>. In this simulation the connectivity strength is set to <inline-formula id="inf156">
<mml:math id="m160">
<mml:mrow>
<mml:mi>&#x3c3;</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1.0</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula> and <inline-formula id="inf157">
<mml:math id="m161">
<mml:mrow>
<mml:msub>
<mml:mi>c</mml:mi>
<mml:mi>o</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>2.0</mml:mn>
<mml:mi>&#x3bc;</mml:mi>
<mml:mi>M</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>. Red line (black line) denotes the adjoining (oblique) cluster configurations of size <inline-formula id="inf158">
<mml:math id="m162">
<mml:mrow>
<mml:mn>10</mml:mn>
<mml:mo>&#xd7;</mml:mo>
<mml:mn>10</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>. Here, <inline-formula id="inf159">
<mml:math id="m163">
<mml:mrow>
<mml:mi>T</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> is computed by averaging over 50 independent simulations, and the error bars are computed as the standard deviation 5 sets of 10 simulation runs each.</p>
</caption>
<graphic xlink:href="fphys-16-1731863-g005.tif">
<alt-text content-type="machine-generated">Line graph showing the relationship between T (in milliseconds) on the y-axis and p on the x-axis. It features two data sets with error bars: a red line with square markers increasing steeply and a black line with circular markers increasing steadily. Both lines exhibit upward trends.</alt-text>
</graphic>
</fig>
<p>Experimental studies reveal that RyR2 clusters within the junctional sarcoplasmic reticulum exhibit highly heterogeneous spatial arrangements. To investigate how this structural diversity affects functional properties, we have also developed a preferential attachment growth model that generates realistic cluster morphologies. The preferential attachment algorithm sequentially places RyR2 channels on a two-dimensional lattice. For each new channel placement, the probability of selecting an empty site <inline-formula id="inf160">
<mml:math id="m164">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>i</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>j</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> is proportional to <inline-formula id="inf161">
<mml:math id="m165">
<mml:mrow>
<mml:msup>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mn>1</mml:mn>
<mml:mo>&#x2b;</mml:mo>
<mml:msub>
<mml:mi>n</mml:mi>
<mml:mi>e</mml:mi>
</mml:msub>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mi>&#x3b1;</mml:mi>
</mml:msup>
<mml:mo>,</mml:mo>
</mml:mrow>
</mml:math>
</inline-formula> where <inline-formula id="inf162">
<mml:math id="m166">
<mml:mrow>
<mml:msub>
<mml:mi>n</mml:mi>
<mml:mi>e</mml:mi>
</mml:msub>
</mml:mrow>
</mml:math>
</inline-formula> represents the number of occupied nearest-neighbor sites and the exponent <inline-formula id="inf163">
<mml:math id="m167">
<mml:mrow>
<mml:mi>&#x3b1;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> is referred to as the clustering parameter. This formulation ensures that sites adjacent to existing channels are preferentially selected, with the strength of this preference controlled by <inline-formula id="inf164">
<mml:math id="m168">
<mml:mrow>
<mml:mi>&#x3b1;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>. When <inline-formula id="inf165">
<mml:math id="m169">
<mml:mrow>
<mml:mi>&#x3b1;</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>0</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>, channel placement is uniform and random, while increasing values promote increasingly compact cluster formation.</p>
<p>
<xref ref-type="fig" rid="F6">Figure 6A</xref> demonstrates the range of cluster structures generated by varying the clustering parameter <inline-formula id="inf166">
<mml:math id="m170">
<mml:mrow>
<mml:mi>&#x3b1;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>. At <inline-formula id="inf167">
<mml:math id="m171">
<mml:mrow>
<mml:mi>&#x3b1;</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>0</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>, RyR2 channels are scattered randomly on the grid. At intermediate values <inline-formula id="inf168">
<mml:math id="m172">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3b1;</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>5</mml:mn>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> more cohesive structures start to from, while retaining some fragmentation. A high clustering parameter <inline-formula id="inf169">
<mml:math id="m173">
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3b1;</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>8</mml:mn>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
</mml:math>
</inline-formula> generates compact arrangements with a few dispersed channels. To examine how cluster structural heterogeneity affects spontaneous Ca spark dynamics, we applied the preferential attachment algorithm by placing 50 RyR2 channels on a 20 &#xd7; 20 lattice domain. Given that individual RyR2 channels are approximately <inline-formula id="inf170">
<mml:math id="m174">
<mml:mrow>
<mml:mn>30</mml:mn>
<mml:mi>n</mml:mi>
<mml:mi>m</mml:mi>
<mml:mo>&#xd7;</mml:mo>
<mml:mn>30</mml:mn>
<mml:mi>n</mml:mi>
<mml:mi>m</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> in size (<xref ref-type="bibr" rid="B7">Cabra et al., 2016</xref>), this domain corresponds to approximately <inline-formula id="inf171">
<mml:math id="m175">
<mml:mrow>
<mml:mn>600</mml:mn>
<mml:mi>n</mml:mi>
<mml:mi>m</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> in diameter, roughly the spatial extent over which Ca is strongly diffusively coupled. For each value of the clustering parameter <inline-formula id="inf172">
<mml:math id="m176">
<mml:mrow>
<mml:mi>&#x3b1;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>, we generated 50 independent cluster configurations. Once the configuration is generated the channels are coupled according to the oblique architecture. We then average the mean first passage time <inline-formula id="inf173">
<mml:math id="m177">
<mml:mrow>
<mml:mi>T</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> over all configurations. <xref ref-type="fig" rid="F6">Figure 6B</xref> shows <inline-formula id="inf174">
<mml:math id="m178">
<mml:mrow>
<mml:mi>T</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> vs. <inline-formula id="inf175">
<mml:math id="m179">
<mml:mrow>
<mml:mi>&#x3b1;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> at a fixed <inline-formula id="inf176">
<mml:math id="m180">
<mml:mrow>
<mml:msub>
<mml:mi>c</mml:mi>
<mml:mi>o</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>2.5</mml:mn>
<mml:mi>&#x3bc;</mml:mi>
<mml:mi>M</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>. The results show that <inline-formula id="inf177">
<mml:math id="m181">
<mml:mrow>
<mml:mi>T</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> increases substantially as <inline-formula id="inf178">
<mml:math id="m182">
<mml:mrow>
<mml:mi>&#x3b1;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> is varied in the range 0&#x2013;10. At low values (<inline-formula id="inf179">
<mml:math id="m183">
<mml:mrow>
<mml:mi>&#x3b1;</mml:mi>
<mml:mo>&#x2248;</mml:mo>
<mml:mn>0</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>), corresponding to dispersed clusters with weak inter-channel connectivity, the mean waiting time is approximately <inline-formula id="inf180">
<mml:math id="m184">
<mml:mrow>
<mml:mn>5</mml:mn>
<mml:mo>&#xd7;</mml:mo>
<mml:msup>
<mml:mn>10</mml:mn>
<mml:mn>3</mml:mn>
</mml:msup>
<mml:mi>m</mml:mi>
<mml:mi>s</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>. As <inline-formula id="inf181">
<mml:math id="m185">
<mml:mrow>
<mml:mi>&#x3b1;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> increases, producing tightly organized clusters, <inline-formula id="inf182">
<mml:math id="m186">
<mml:mrow>
<mml:mi>T</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> increases to approximately <inline-formula id="inf183">
<mml:math id="m187">
<mml:mrow>
<mml:msup>
<mml:mn>10</mml:mn>
<mml:mn>5</mml:mn>
</mml:msup>
<mml:mi>m</mml:mi>
<mml:mi>s</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> for the oblique configuration, and <inline-formula id="inf184">
<mml:math id="m188">
<mml:mrow>
<mml:msup>
<mml:mn>10</mml:mn>
<mml:mn>6</mml:mn>
</mml:msup>
</mml:mrow>
</mml:math>
</inline-formula> for the adjoining configuration. This effect saturates above <inline-formula id="inf185">
<mml:math id="m189">
<mml:mrow>
<mml:mi>&#x3b1;</mml:mi>
<mml:mo>&#x2248;</mml:mo>
<mml:mn>7</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula> since all clusters generated for larger <inline-formula id="inf186">
<mml:math id="m190">
<mml:mrow>
<mml:mi>&#x3b1;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> are close to fully compact and yield the same <inline-formula id="inf187">
<mml:math id="m191">
<mml:mrow>
<mml:mi>T</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> as the fully connected square lattice. This exponential relationship, in the range 0&#x2013;7, demonstrates that cluster compactness is a critical determinant of spontaneous Ca release dynamics, with tightly organized structures providing exponentially greater stability during diastole.</p>
<fig id="F6" position="float">
<label>FIGURE 6</label>
<caption>
<p>Spark timing in heterogeneous clusters. <bold>(A)</bold> Representative cluster morphologies generated by varying the clustering parameter <inline-formula id="inf188">
<mml:math id="m192">
<mml:mrow>
<mml:mi>&#x3b1;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> in the preferential attachment algorithm. Low values yield dispersed clusters with irregular boundaries, intermediate values produce partially cohesive structures, and high values generate compact, dense aggregates. In this simulation we place <inline-formula id="inf189">
<mml:math id="m193">
<mml:mrow>
<mml:mn>50</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula> RyR2 channels on a grid of size <inline-formula id="inf190">
<mml:math id="m194">
<mml:mrow>
<mml:mn>20</mml:mn>
<mml:mo>&#xd7;</mml:mo>
<mml:mn>20</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula>. Neighboring channels are connected according to the oblique geometry. <bold>(B)</bold> Dependence of the mean waiting time <inline-formula id="inf191">
<mml:math id="m195">
<mml:mrow>
<mml:mi>T</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> for spontaneous Ca spark initiation on the clustering parameter. Red and black line corresponds to <inline-formula id="inf192">
<mml:math id="m196">
<mml:mrow>
<mml:mi>&#x3c3;</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>1</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula> and <inline-formula id="inf193">
<mml:math id="m197">
<mml:mrow>
<mml:mi>&#x3c3;</mml:mi>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>0.5</mml:mn>
</mml:mrow>
</mml:math>
</inline-formula> respectively. The waiting time <inline-formula id="inf194">
<mml:math id="m198">
<mml:mrow>
<mml:mi>T</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> is computed by averaging over 50 independent simulation runs. For each simulation run a different cluster arrangement is generated using the preferential growth algorithm. Here, we set <inline-formula id="inf195">
<mml:math id="m199">
<mml:mrow>
<mml:msub>
<mml:mi>c</mml:mi>
<mml:mi>o</mml:mi>
</mml:msub>
<mml:mo>&#x3d;</mml:mo>
<mml:mn>2.5</mml:mn>
<mml:mi>&#x3bc;</mml:mi>
<mml:mi>M</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>, and error bars are computed using 5 sets of 10 simulation runs.</p>
</caption>
<graphic xlink:href="fphys-16-1731863-g006.tif">
<alt-text content-type="machine-generated">Panel A shows three grid patterns with varying red clusters at &#x3B1; values of zero, five, and eight. Panel B displays a line graph with two plots showing the relationship between &#x3B1; and time (T) in milliseconds. The red line increases sharply compared to the slower increase of the black line.</alt-text>
</graphic>
</fig>
</sec>
</sec>
<sec sec-type="discussion" id="s4">
<title>Discussion</title>
<sec id="s4-1">
<title>Inter-subunit coupling controls cluster response to Ca</title>
<p>In this study, we developed a computational model of an RyR2 cluster in which each channel is composed of four interacting subunits, with channels arranged in a configuration consistent with structures observed in cryo-EM and super-resolution imaging. Our goal was to understand how inter-subunit cooperativity influences Ca leak in cardiac cells. The first result of the model is that the cluster exhibits a sharp contrast in behavior across different Ca concentration regimes. At Ca concentrations larger than <inline-formula id="inf196">
<mml:math id="m200">
<mml:mrow>
<mml:mn>100</mml:mn>
<mml:mi>&#x3bc;</mml:mi>
<mml:mi>M</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> RyR2 clusters respond rapidly to Ca signals on sub-millisecond timescales. This fast response is largely insensitive to the strength of channel-to-channel coupling. However, at lower Ca concentrations (below <inline-formula id="inf197">
<mml:math id="m201">
<mml:mrow>
<mml:mo>&#x223c;</mml:mo>
<mml:mn>5</mml:mn>
<mml:mi>&#x3bc;</mml:mi>
<mml:mi>M</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>), the system behaves very differently. In this case, the mean waiting time to a spontaneous Ca spark becomes exponentially sensitive to the strength of subunit coupling. Specifically, at a Ca concentration of <inline-formula id="inf198">
<mml:math id="m202">
<mml:mrow>
<mml:mn>5.0</mml:mn>
<mml:mi>&#x3bc;</mml:mi>
<mml:mi>M</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>, increasing the coupling strength <inline-formula id="inf199">
<mml:math id="m203">
<mml:mrow>
<mml:mi>&#x3c3;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> between RyR2 subunits from 0 to 0.5 (for the adjoining arrangement), leads to an almost 4 orders of magnitude increase in the mean waiting time. This behavior highlights the functional significance of inter-subunit coupling: it preserves fast responsiveness to LCC openings while ensuring cluster stability at lower Ca concentrations. An important consequence of this finding is that diastolic Ca leak, which is determined by how often spontaneous Ca sparks occur, will be exponentially sensitive to the coupling between RyR2 within clusters. Therefore, even small changes in coupling strength can produce orders-of-magnitude differences in leak rate. These results highlights the critical role of RyR2 coupling in regulating Ca cycling homeostasis.</p>
</sec>
<sec id="s4-2">
<title>Ca leak is exponentially sensitive to RyR2 cluster integrity</title>
<p>Experimental evidence from cryo-EM and super-resolution microscopy shows that RyR2 clusters in cardiac cells are highly irregular, forming heterogeneous structures rather than perfect lattices. To examine how this heterogeneity shapes Ca signaling, we extended our modeling framework to include disordered cluster geometries and systematically varied their structural integrity. In one approach, we introduced a coupling parameter <inline-formula id="inf200">
<mml:math id="m204">
<mml:mrow>
<mml:mi>p</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>, defined as the probability that a subunit remains linked to its nearest neighbor. This parameter represents the progressive loss of inter-subunit connectivity. As <inline-formula id="inf201">
<mml:math id="m205">
<mml:mrow>
<mml:mi>p</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> decreased from 1 (fully intact) to 0 (fully fragmented), the mean waiting time to a spontaneous Ca spark shortened by approximately 3 orders of magnitude. These results shows that Ca spark frequency is exponentially sensitive to even small disruptions in cluster connectivity. In parallel, we used a preferential attachment algorithm to generate clusters with heterogeneous shapes resembling those observed experimentally. Here, the clustering parameter <inline-formula id="inf202">
<mml:math id="m206">
<mml:mrow>
<mml:mi>&#x3b1;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> controlled the likelihood that new channels attached to existing groups, thereby tuning the degree of structural cohesion. Increasing <inline-formula id="inf203">
<mml:math id="m207">
<mml:mrow>
<mml:mi>&#x3b1;</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> produced a substantial increase in spark waiting time, demonstrating that more tightly connected clusters are exponentially more stable at lower Ca concentrations. Together, these results reveal that heterogeneity in RyR2 cluster organization plays a central role in regulating the timing of spontaneous Ca sparks.</p>
</sec>
<sec id="s4-3">
<title>Ca leak in heart failure is dependent on RyR2 structural integrity</title>
<p>Recent high-resolution imaging studies have revealed that RyR2 clusters lose their structural integrity in heart failure, with important functional consequences for Ca handling. <xref ref-type="bibr" rid="B31">Sheard et al. (2022)</xref> used enhanced expansion microscopy to demonstrate that RyR2 clusters in failing myocytes frequently exhibit a frayed appearance, where small groups of channels detach from the main cluster, particularly in regions depleted of the structural protein junctophilin-2. Similarly, <xref ref-type="bibr" rid="B21">Kolstad et al. (2018)</xref> showed that post-infarction heart failure is characterized by RyR2 cluster dispersion and fragmentation, resulting in smaller, more numerous cluster fragments with reduced inter-channel connectivity. Our computational modelling provides a mechanistic explanation for why these structural changes change the Ca leak rate. The key insight from our model is that spontaneous Ca spark frequency exhibits exponential sensitivity to cluster structural integrity. Mechanistically, intact inter-subunit coupling within the cluster provides stabilizing interactions that maintain channels in the closed state during diastole through cooperative inhibition. When coupling is weakened by fragmentation, this stabilizing effect is lost, making the cluster far more susceptible to stochastic activation. The key new insight in this study is that this relationship is exponential, and explains why even modest cluster remodeling, such as partial fragmentation or fraying, can produce the orders of magnitude increases in diastolic Ca leak observed in heart failure. Thus, cluster architectural integrity emerges as a critical control mechanism that drives Ca leak.</p>
</sec>
<sec id="s4-4">
<title>The role of diffusion in RyR2 cluster dynamics</title>
<p>Ca diffusion plays a crucial role in determining the functional coupling between RyR2 channels within the dyadic cleft. With a diffusion coefficient of approximately <inline-formula id="inf204">
<mml:math id="m208">
<mml:mrow>
<mml:mn>100</mml:mn>
<mml:mo>&#x2212;</mml:mo>
<mml:mn>500</mml:mn>
<mml:mtext>&#x2009;</mml:mtext>
<mml:msup>
<mml:mrow>
<mml:mfenced open="(" close=")" separators="&#x7c;">
<mml:mrow>
<mml:mi>&#x3bc;</mml:mi>
<mml:mi>m</mml:mi>
</mml:mrow>
</mml:mfenced>
</mml:mrow>
<mml:mn>2</mml:mn>
</mml:msup>
<mml:mo>/</mml:mo>
<mml:mi>s</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> and an RyR2 open time of roughly <inline-formula id="inf205">
<mml:math id="m209">
<mml:mrow>
<mml:mn>1</mml:mn>
<mml:mi>m</mml:mi>
<mml:mi>s</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>, Ca diffuses roughly <inline-formula id="inf206">
<mml:math id="m210">
<mml:mrow>
<mml:mn>600</mml:mn>
<mml:mi>n</mml:mi>
<mml:mi>m</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula> before channel closure. Since this diffusion length exceeds the diameter of even the largest RyR2 clusters (<inline-formula id="inf207">
<mml:math id="m211">
<mml:mrow>
<mml:mo>&#x2264;</mml:mo>
<mml:mn>400</mml:mn>
<mml:mi>n</mml:mi>
<mml:mi>m</mml:mi>
</mml:mrow>
</mml:math>
</inline-formula>), Ca released from any open channel rapidly equilibrates across the entire dyadic space, creating strong functional coupling through the shared Ca microdomain. Importantly, when clusters become frayed or fragmented into separate pieces, the disjoint subclusters often remain within the same Ca microdomain and are therefore still functionally coupled through diffusion. However, these separated channels lose the stabilizing nearest-neighbor interactions that maintain closure during diastole. A small detached subcluster can thus spontaneously open without the energetic penalty of intact inter-channel coupling yet still elevate local Ca sufficiently to recruit the larger cluster. This asymmetry between lost stabilization and preserved diffusive coupling explains why frayed clusters exhibit exponentially faster spontaneous Ca spark frequencies.</p>
</sec>
<sec id="s4-5">
<title>Cluster integrity as a therapeutic target</title>
<p>The exponential sensitivity of Ca leak to RyR2 coupling revealed in our study carries significant therapeutic implications. Because spark frequency rises exponentially as coupling weakens, even subtle structural disruptions can translate into exponentially large increases in spontaneous Ca release frequency. This nonlinearity also works in the opposite direction, so that modest improvements in cluster integrity could produce dramatic reductions in Ca leak. These results shows that structural stability itself is a valuable therapeutic target. Strategies that preserve or restore RyR2 subunit coupling, such as enhancing the stabilizing action of accessory proteins like FKBP12.6 (<xref ref-type="bibr" rid="B37">Wehrens et al., 2003</xref>), offer a direct way to reinforce this stability. Other proteins that regulate RyR2 positioning or inter-channel contacts could likewise be harnessed to maintain cluster integrity (<xref ref-type="bibr" rid="B29">Reynol et al., 2016</xref>). Such approaches differ fundamentally from traditional interventions aimed at modifying channel gating kinetics or expression levels, which may have limited efficacy if the underlying cluster architecture remains compromised. By identifying inter-subunit coupling as a central determinant of spontaneous Ca release, our study highlights a specific and tunable structural feature of the RyR2 complex that could be targeted to suppress pathological Ca leak.</p>
</sec>
<sec id="s4-6">
<title>Model limitations</title>
<p>Our model has several important limitations. First, we do not incorporate the complex regulatory mechanisms known to modulate RyR2 gating in cardiac myocytes, including inhibition by physiological Mg concentrations and modulation by regulatory proteins such as CaMKII, calmodulin, and FKBP12.6 (<xref ref-type="bibr" rid="B26">Marx and Marks, 2013</xref>; <xref ref-type="bibr" rid="B36">Walweel and Laver, 2015</xref>; <xref ref-type="bibr" rid="B40">Zahradn&#xed;kov&#xe1; et al., 2025</xref>; <xref ref-type="bibr" rid="B30">Rokita and Anderson, 2012</xref>). These factors will alter the spark waiting times. However, they are unlikely to eliminate the exponential dependence on cluster connectivity that emerges from cooperative gating between channels. Second, we model RyR2 clusters using idealized geometric arrangements of either pure oblique or pure adjoining configurations, whereas native clusters in cardiac cells exhibit heterogeneous combinations of both interaction types. Our results indicate that mixed geometries will yield intermediate waiting times, with the precise value depending on the relative proportion of each interaction type. However, because both pure configurations exhibit exponential sensitivity to coupling strength and connectivity, we expect this relationship to persist regardless of geometric details. Third, our framework does not include channel inactivation, SR Ca depletion, or competing Ca clearance mechanisms such as the sodium calcium exchanger and the sarcoplasmic reticulum Ca ATPase (SERCA). These processes determine spark termination and recovery. However, they occur on timescales much shorter than the diastolic waiting times we investigate and therefore do not affect the spark initiation dynamics that are the focus of this study. While future work incorporating more detailed RyR2 kinetics and realistic cluster geometries will refine quantitative predictions, the central finding that spontaneous Ca spark frequency is exponentially sensitive to cluster structural integrity follows directly from the cooperativity between RyR2 channels and should remain robust.</p>
</sec>
</sec>
</body>
<back>
<sec sec-type="data-availability" id="s5">
<title>Data availability statement</title>
<p>The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.</p>
</sec>
<sec sec-type="author-contributions" id="s6">
<title>Author contributions</title>
<p>AN: Conceptualization, Data curation, Formal Analysis, Investigation, Methodology, Software, Validation, Visualization, Writing &#x2013; original draft, Writing &#x2013; review and editing. YS: Conceptualization, Methodology, Project administration, Validation, Writing &#x2013; original draft, Writing &#x2013; review and editing, Investigation, Software.</p>
</sec>
<ack>
<title>Acknowledgements</title>
<p>We thank these agencies for their generous support.</p>
</ack>
<sec sec-type="COI-statement" id="s8">
<title>Conflict of interest</title>
<p>The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.</p>
</sec>
<sec sec-type="ai-statement" id="s9">
<title>Generative AI statement</title>
<p>The authors declare that Generative AI was used in the creation of this manuscript. Generative AI was used to check grammar and sentence construction.</p>
<p>Any alternative text (alt text) provided alongside figures in this article has been generated by Frontiers with the support of artificial intelligence and reasonable efforts have been made to ensure accuracy, including review by the authors wherever possible. If you identify any issues, please contact us.</p>
</sec>
<sec sec-type="disclaimer" id="s10">
<title>Publisher&#x2019;s note</title>
<p>All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.</p>
</sec>
<fn-group>
<fn fn-type="custom" custom-type="edited-by">
<p>
<bold>Edited by:</bold> <ext-link ext-link-type="uri" xlink:href="https://loop.frontiersin.org/people/730341/overview">Sean Michael Wilson</ext-link>, Loma Linda University, United States</p>
</fn>
<fn fn-type="custom" custom-type="reviewed-by">
<p>
<bold>Reviewed by:</bold> <ext-link ext-link-type="uri" xlink:href="https://loop.frontiersin.org/people/36338/overview">D. George Stephenson</ext-link>, La Trobe University, Australia</p>
<p>
<ext-link ext-link-type="uri" xlink:href="https://loop.frontiersin.org/people/3271345/overview">Jun He</ext-link>, Nanjing Normal University, China</p>
</fn>
</fn-group>
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