On the Formation and Breakup of Spiral Waves of Calcium

Model Status

This model contains partial differentials and as such can not currently be solved by existing CellML tools.

Model Structure

In many cell types, Ca²⁺ acting as a signalling agent plays an important role in the control of cellular function. It is thought that these signals are mediated through oscillations in intracellular Ca²⁺ concentration ([Ca²⁺]_i). The mechanisms underlying these oscillations are complex and they have only been partially elucidated. In addition, in larger cells such as in the Xenopus laevis oocyte, a variety of Ca²⁺ wave behaviour has been seen, including plane waves, pulsating patterns and spiral waves. This has lead to the frequent use of Xenopus oocytes in experimental studies of Ca²⁺ waves.

The probability of the inositol 1,4,5-triphosphate (IP₃) receptor being in the open state is influenced by both IP₃ and Ca²⁺. This duel system of control is thought to underlie Ca²⁺ waves and oscillations. IP₃ binding to the receptor activates the channel, causing a small amount of Ca²⁺ to diffuse out of the ER into the cytosol. This Ca²⁺ has a positive feedback effect, called calcium-induced calcium release (CICR), and it is thought to underlie Ca²⁺ wave propagation. At intermediate IP₃ concentrations, a stable limit cycle appears in the kinetics, and the model becomes oscillatory. Under these conditions, wave propagation can occur even without excitability.

Several mathematical models for Ca²⁺ wave propagation have been published (for some examples see DeYoung and Keizer's IP₃-mediated Ca²⁺ release model, 1992 and Li and Rinzel's IP₃-mediated Ca²⁺ oscillations model, 1994), all of which are similar. The mathematical model described here was published by McKenzie and Sneyd in 1998 (see the figure below). It describes the process by which IP₃ diffuses through the cytosol and binds to receptors located on the endoplasmic reticulum (ER). These receptors also function as Ca²⁺ channels. They open upon IP₃ binding, and Ca²⁺ diffuses out of the ER down its electrochemical gradient into the cytosol. Ca²⁺ is pumped against its electrochemical gradient out of the cytosol into the ER, and ions also leak into the cytosol from the ER and the external environment as they diffuse into the cytosol down their electrochemical gradient.

Model simulations showed that spiral waves could be induced by simulating the release of IP₃. Increasing the size of the IP₃ additions caused a decrease in the rotation period and the breakup of the spiral wave solutions. Following this, irregular spatio-temporal patterns occurred. From their model simulations, the authors predict that a simple experimental procedure may be sufficient to reproduce unstable spiral waves.

The complete original paper reference is cited below:

On the Formation and Breakup of Spiral Waves of Calcium, A. McKenzie and J. Sneyd, 1998, International Journal of Bifurcation and Chaos, 8, 2003-2012.(A PDF version of the article is available on James Sneyd's website.)

The raw CellML descriptions of the model can be downloaded in various formats as described in .

A schematic diagram representing the model.