A bidomain threshold model of propagating calcium waves

Abstract

We present a bidomain fire-diffuse-fire model that facilitates mathematical analysis of propagating waves of elevated intracellular calcium (Ca 2+ ) in living cells. Modelling Ca 2+ release as a threshold process allows the explicit construction of travelling wave solutions to probe the dependence of Ca 2+ wave speed on physiologically important parameters such as the threshold for Ca 2+ release from the endoplasmic reticulum (ER) to the cytosol, the rate of Ca 2+ resequestration from the cytosol to the ER, and the total [Ca 2+ ] (cytosolic plus ER). Interestingly, linear stability analysis of the bidomain fire-diffuse-fire model predicts the onset of dynamic wave instabilities leading to the emergence of Ca 2+ waves that propagate in a back-and-forth manner. Numerical simulations are used to confirm the presence of these so-called 'tango waves' and the dependence of Ca 2+ wave speed on the total [Ca 2+ ].

Sensitisation waves in a bidomain fire-diffuse-fire model of intracellular Ca2+ dynamics

“…Computing the wave profile in closed form, we probe the dependence of the wave speed on physiologically relevant parameters. Moreover, a linear stability analysis reveals the possibility of dynamic instabilities that give rise to Tango waves [27,28]. We support all analytical results with direct numerical simulations.…”

“…Figure 1 illustrates how varying the strength of Ca 2+ sequestration alters the wave speed of a travelling pulse for various values of κ. We re-scaled both time scales, τ and τ er , by the same factor r, so that the total Ca 2+ concentration c ∞ T in the wake and in the front of the pulse remains unchanged [28]. Thus, an acute decrease in SERCA activity as performed experimentally in [10] corresponds to decreasing r in Fig.…”

“…These single domain threshold models treat the ER/SR as an infinite store so that release events do not change the luminal Ca 2+ concentration. Following recent work [27,28], we here employ a bidomain threshold model that incorporates dynamics in both the cytosol and the lumen:…”

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

“…Early works focused on a one-dimensional representation of the cytosol and were instrumental in our understanding of calcium waves in cardiac myocytes (Dupont and Goldbeter 1994, Keizer et al 1998, Dawson et al 1999. More recently, these models were extended to incorporate stochastic release and a dynamic SR (Coombes and Timofeeva 2003, Coombes et al 2004, Thul et al 2008. 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.…”

Translating Intracellular Calcium Signaling into Models

“…Instead of running a large number of simulations all that is needed is to evaluate analytical expressions, which can be done in a fraction of the time that is required for the numerical simulations. For example, the impact of SERCA pumps on Ca 2+ wave propagation has been studied in (Coombes 2001), and investigating the interplay between the cytosol and the SR provided explanations for two novel wave types: tango waves (Li 2003;Thul et al 2008b) and sensitization waves (Keller et al 2007;Thul et al 2009a). While deterministic models such as the original FDF description are still instrumental in advancing our understanding of intracellular Ca 2+ waves, hybrid frameworks that incorporate the stochastic nature of Ca 2+ release have gained considerable attention.…”