Rationale: Heart muscle contraction is  activated by a synchronized systolic Ca release from sarcoplasmic reticulum (SR) via Ca sparks. In disease, Ca sparks fail to terminate, causing a diastolic Ca leak that decreases contraction amplitude and increases the risk of arrhythmia. The mechanisms and treatment of the abnormal Ca leak remain unclear. We have recently shown that spark termination emerges as collective behavior (synchronized closings) of Ca release channels (RyRs) that is identical to synchronization of spin orientation in ferromagnets described by a phase transition in Ising model . Objective: We employed the Ising approach to investigate and classify mechanisms of spark termination failure and Ca leak. Methods and Results: The key parametersdetermining whether spark termination succeeds or fails are SR Ca and RyR opening/closing rates. They define analogues of magnetic field h and the inverse temperature (\beta) in Ising model\beta*\beta* \beta*  . Sparks terminate via a phase transition known as "hpolarity reversal" and the leak emerges when hfails to change its sign. This happens when the SR depletes insufficiently and RyR openings remained partially synchronized via Ca-induced-Ca-release, generating long-lasting sparks. Leak can also occur via \beta, known as Onsager's "order-disorder" transition. This happens at low SR Ca, reducing RyR current and RyRs interactions, resulting in independent RyR openings. The disorder leak is distinguished from synchronized leak  by larger Peierls contour lengths  reflecting degree of disorder. Conclusions: Abnormal leak results from either a probability imbalance during synchronized firing of RyRs or from disordered RyR firing. Each leak type requires different and balanced treatment to shift RyR operation towards normal spark termination .
 Heart muscle contraction is normally activated by a synchronized Ca release from sarcoplasmic reticulum (SR) , a major intracellular Ca store. However, under abnormal conditions Ca leaks from the SR, decreasing heart contraction amplitude and increasing risk of life-threatening arrhythmia. The mechanisms and regimes of SR operation generating the abnormal Ca leak remain unclear. Here we employed both numerical and analytical modeling to get mechanistic insights into the emergent Ca leak phenomenon. Our numerical simulations using a detailed realistic model of Ca release unit (CRU) reveal sharp transitions resulting in Ca leak. The emergence of leak is closely mapped mathematically to the Ising model from statistical mechanics. The system steady-state behavior is determined by two aggregate parameters: the analogues of magnetic field (h) and the inverse temperature (\beta) in the Ising model, for which we have explicit formulas in terms of SR Ca and release channel opening/closing rates. The classification of leak regimes takes the shape of a phase \beta-h diagram, with the regime boundaries occurring at h=0 and a critical value of \beta (\beta*) which we estimate using a classical Ising model and mean field theory. Our theory predicts that a synchronized Ca leak will occur when h>0 and \beta>\beta* and a disordered leak occurs when \beta<\beta* and h is not too negative . The disorder leak is distinguished from synchronized leak (in long-lasting sparks) by larger Peierls contour lengths , an output parameter reflecting degree of disorder. Thus, in addition to our detailed numerical model approach we also offer an instantaneous computational tool using analytical formulas of the Ising model for respective RyR parameters and SR Ca load that describe and classify phase transitions and leak emergence .
D
Rationale:
A
normally
D
systolic
R
via Ca sparks. In disease, Ca sparks fail to terminate, causing a diastolic Ca leak that decreases
, a major intracellular Ca store. However, under abnormal conditions Ca leaks from the SR, decreasing heart
R
increases the risk of
increasing risk of life-threatening
R
treatment of
regimes of SR operation generating
R
We have recently shown that spark termination emerges as collective behavior (synchronized closings)
Here we employed both numerical and analytical modeling to get mechanistic insights into the emergent Ca leak phenomenon. Our numerical simulations using a detailed realistic model
R
channels (RyRs) that is identical to synchronization of spin orientation in ferromagnets described by a phase transition in Ising model . Objective: We employed the Ising approach to investigate and classify mechanisms of spark termination failure and Ca leak. Methods and Results: The key parametersdetermining whether spark termination succeeds or fails are SR Ca and RyR opening/closing rates. They define
unit (CRU) reveal sharp transitions resulting in Ca leak. The emergence of leak is closely mapped mathematically to the Ising model from statistical mechanics. The system steady-state behavior is determined by two aggregate parameters: the
R
h
(h)
R
Ising model
the Ising model, for which we have explicit formulas in terms of SR Ca and release channel opening/closing rates. The classification of leak regimes takes the shape of a phase \beta-h diagram, with the regime boundaries occurring at h=0 and a critical value of \beta (
A
) which we estimate using a classical Ising model and mean field theory. Our theory predicts that a synchronized Ca leak will occur when h>0 and \beta>
A
and a disordered leak occurs when \beta<
A
and h is not too negative
D
Sparks terminate via a phase transition known as "hpolarity reversal" and the leak emerges when hfails to change its sign. This happens when the SR depletes insufficiently and RyR openings remained partially synchronized via Ca-induced-Ca-release, generating long-lasting sparks. Leak can also occur via \beta, known as Onsager's "order-disorder" transition. This happens at low SR Ca, reducing RyR current and RyRs interactions, resulting in independent RyR openings.
A
(in long-lasting sparks)
A
, an output parameter
R
Conclusions: Abnormal leak results from either a probability imbalance during synchronized firing of RyRs or from disordered RyR firing. Each leak type requires different and balanced treatment to shift RyR operation towards normal spark termination
Thus, in addition to our detailed numerical model approach we also offer an instantaneous computational tool using analytical formulas of the Ising model for respective RyR parameters and SR Ca load that describe and classify phase transitions and leak emergence