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. 2023 Sep 28;134(12):124701.
doi: 10.1063/5.0151255. Epub 2023 Sep 22.

Elementary intracellular Ca signals approximated as a transition of release channel system from a metastable state

Guillermo Veron  1 , Victor A Maltsev  1 , Michael D Stern  1 , Anna V Maltsev  2
Affiliations

Elementary intracellular Ca signals approximated as a transition of release channel system from a metastable state

Guillermo Veron et al. J Appl Phys. .

Abstract

Cardiac muscle contraction is initiated by an elementary Ca signal (called Ca spark) which is achieved by collective action of Ca release channels in a cluster. The mechanism of this synchronization remains uncertain. We approached Ca spark activation as an emergent phenomenon of an interactive system of release channels. We constructed a weakly lumped Markov chain that applies an Ising model formalism to such release channel clusters and probable open channel configurations and demonstrated that spark activation is described as a system transition from a metastable to an absorbing state, analogous to the pressure required to overcome surface tension in bubble formation. This yielded quantitative estimates of the spark generation probability as a function of various system parameters. We performed numerical simulations to find spark probabilities as a function of sarcoplasmic reticulum Ca concentration, obtaining similar values for spark activation threshold as our analytic model, as well as those reported in experimental studies. Our parametric sensitivity analyses also showed that the spark activation threshold decreased as Ca sensitivity of RyR activation and RyR cluster size increased.

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Conflict of interest statement

The authors have no conflicts to disclose.

Figures

FIG. 1.
FIG. 1.
The definition and numerical simulations of the RyR interaction profile ψ(r) that determines Ca-induced-Ca release in our model. ψ(r) is a steady-state [Ca] in the dyadic cleft as a function of distance r when one RyR opens at r = 0. The plot shows a family of simulated interaction profiles ψ(r) at various fixed SR Ca loadings from 25 to 1000 μM (the right column shows lines and symbols for each curve). Larger [Ca] at the nearest closed channel at higher SR loading indicates stronger channel interactions and stronger Ca-induced-Ca release. The interaction profiles are measured in numerical simulations of sparks as an instantaneous [Ca] in dyadic space 10 ms after the first channel opens for 9 × 9 RyR grid, the distance between RyRs is 30 nm, and each voxel is 10 × 10 × 15 nm3 in xyz. All other channels are forced to stay closed. See Excel file in the supplementary material for exact values of the profiles that are used in our simulations of the analytical model. Insets (modified from Ref. 17) show the RyR grid and its location with respect to SR, cytoplasm, and cell membrane in our CRU model. Please note that L-type channels are not included in our model of spark activation.
FIG. 2.
FIG. 2.
Our model includes all possible spatial configurations of RyR openings during the initial interaction steps of spark activation after one channel is open acting as a nucleation site. (a) Schematical illustration of five-state Markov process simulating spark evolution in our weakly lumped model. Each arrow represents the event of the Markov process changing from one state to another state with the direction indicated by the arrow. Each black circle shows all possible configurations of open RyRs, independent of how each configuration is reached. (b) Configuration tree. Illustration of all possible configurations and the series of events that could take place to reach each of the configurations. The model has ten possible configurations, including configuration Ø with no open channels. Numbers at each configuration indicate the number of possible ways to reach a given configuration.
FIG. 3.
FIG. 3.
Our analytical and numerical models predict the probability of Ca spark activation as a function of SR Ca loading. (a) The probability of transitioning from two open channels to three open channels (circles) and probabilities of transitioning from three open channels to four open channels via straight configuration (dashed line) or triangle configuration (dotted line). (b) Spark activation predicted numerically and analytically with and analytically without correction for diffusion delay. In the numerical method, the probability of spark firing at each SR Ca is evaluated from 10 000 simulation runs of 200 ms each. In each run, at t = 0, one RyR in the center of a 9 × 9 RyR cluster is set open. Our criterion for spark firing is that 50% of all RyRs open at any moment before all RyRs close. (c), Experimentally defined SR Ca threshold for Ca spark activation; shown are mean values of total spark-mediated release flux (measured by confocal microscopy) which are rescanned and replotted from Fig. 3(b) of Ref. .
FIG. 4.
FIG. 4.
Numerical and analytic models behave essentially the same within a broad range of key model parameters λ and γ. Shown are heatmaps of two-dimensional sensitivity analysis of the SR Ca threshold (SR[Ca]th) for spark initiation with respect to λ and γ in analytical (a) and numerical (b) models; the RyR opening rate is taken to be an exponential of the cleft [Ca] given by λ* exp(γ[Ca]). For these analyses, we set 0.1 probability for spark activation to obtain the associated SR Ca threshold. In turn, in numerical simulations, each threshold is defined from a series of spark activation simulations with increasing SR Ca, and the probability of spark firing at each SR Ca is evaluated from 10 000 simulation runs.
FIG. 5.
FIG. 5.
An example of a numerical model simulation of Ca spark evolution triggered by an opening of one RyR at time = 0 at a random location. (a) Number of open RyRs as a function of time for the entire duration of the spark. (b) RyR openings for the first 2.5 ms. (c) SR Ca depletion during the entire duration of the spark. (d) A minor SR Ca depletion at the moment when four channels open. (e) Detailed spatiotemporal CRU system evolution from one open channel (white arrow) to four open channels in a 9 × 9 RyR grid. The open channel cluster is outlined by a white line. In this example, the spark activation evolves via three transitions, recruiting to fire its neighbors counterclockwise. Channels are shown by green arrows. [Ca] is coded by red shades: 0 μM is pure black and 200 μM is pure red. See more details in Videos S1 and S2 in the supplementary material.
FIG. 6.
FIG. 6.
Numerical model prediction of the probability of Ca spark activation as a function of SR Ca loading for RyR clusters of various sizes. (a) Spark probabilities with fixed SR Ca (dashed lines) vs free-running SR, i.e., SR Ca is not fixed (solid lines). For each data point, probability of spark firing is evaluated from 10 000 simulation runs of 200 ms each. In each run, at t = 0, one RyR in a random location in the respective RyR cluster is set open. (b) SR Ca threshold as a function of the number of RyRs in CRU blue for free-running SR and orange for SR clamp, both fitted with a power function (equations with R2 values are shown in the plots).
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