Evolution of Cardiac Calcium Waves from Stochastic Calcium Sparks

Abstract

We present a model that provides a unified framework for studying Ca2+ sparks and Ca2+ waves in cardiac cells. The model is novel in combining 1) use of large currents (approximately 20 pA) through the Ca2+ release units (CRUs) of the sarcoplasmic reticulum (SR); 2) stochastic Ca2+ release (or firing) of CRUs; 3) discrete, asymmetric distribution of CRUs along the longitudinal (separation distance of 2 microm) and transverse (separated by 0.4-0.8 microm) directions of the cell; and 4) anisotropic diffusion of Ca2+ and fluorescent indicator to study the evolution of Ca2+ waves from Ca2+ sparks. The model mimics the important features of Ca2+ sparks and Ca2+ waves in terms of the spontaneous spark rate, the Ca2+ wave velocity, and the pattern of wave propagation. Importantly, these features are reproduced when using experimentally measured values for the CRU Ca2+ sensitivity (approximately 15 microM). Stochastic control of CRU firing is important because it imposes constraints on the Ca2+ sensitivity of the CRU. Even with moderate (approximately 5 microM) Ca2+ sensitivity the very high spontaneous spark rate triggers numerous Ca2+ waves. In contrast, a single Ca2+ wave with arbitrarily large velocity can exist in a deterministic model when the CRU Ca2+ sensitivity is sufficiently high. The combination of low CRU Ca2+ sensitivity (approximately 15 microM), high cytosolic Ca2+ buffering capacity, and the spatial separation of CRUs help control the inherent instability of SR Ca2+ release. This allows Ca2+ waves to form and propagate given a sufficiently large initiation region, but prevents a single spark or a small group of sparks from triggering a wave.

Cardiac cell modelling: Observations from the heart of the cardiac physiome project

“…While integrating additional elements of calcium signalling into cardiac models has been of ongoing interest, a parallel development has been focusing on the implementation of realistic cellular geometries. This has resulted in three-dimensional simulations of calcium concentration profiles and has provided the first insights into the coupling between cell shape, ion channel and receptor distributions, and calcium signaling (Izu et al 2001, Michailova et al 2002, Li et al 2007, Korhonen et al 2009, Lu et al 2009; and see Lemon 2004 for an analytical study). Although consensus is emerging as regarding core components of any cardiac calcium model, details of some key mechanisms are still debated.…”

Long-Time Simulations on High Resolution Meshes to Model Calcium Waves in a Heart Cell

“…The model of the spark mechanism. The key term of the model is the term J SR (u (0) , x, t) in the equation for the calcium concentration (labeled as species i = 0) in (1.1) that describes the release of calcium at the calcium release units (CRUs), referred to as spark events [11,13]. On the spatial scale of a cell, the CRUs appear as discrete points distributed uniformly throughout the cell.…”

“…The desired effect of this design is that the calcium released at one CRU diffuses to a neighboring CRU, whose probability for opening increases with the increased calcium concentration. If the calcium concentration then reaches a third CRU and it opens, the effect is that of a wave forming throughout the cell [8,13]. After a CRU closes again, it cannot release calcium again for a time period t closed = 100 ms.…”

“…Introduction. Diffusion waves of calcium ions in a heart cell are part of the normal functioning of the heart, but can also trigger arrhythmias (irregular heart beat) [11,12,13]. See these sources as well as the appendix in [8] for more references to background material.…”

“…Specifically, a final time on the order of 1,000 ms is our goal, because it will allow enough time for waves to self-organize and for several waves of calcium to traverse the cell, given their typical time scale of 100 ms; see section 2.1. We use a domain with one long dimension of 64 μm and a cross section of 12.8 × 12.8 μm 2 , which is a good representation of the scale and the fundamental shape of a cell in this context [11,13]. A particular example of quantities under still-active validation is the positioning of and distances between the release units of calcium; see, e.g., [2] and [12] that approach the problem from the experimental and computational side, respectively.…”