ORIGINAL PAPER Ca 2+ spiral waves in a spatially discrete and random medium Jun Tang Æ Lijian Yang Æ Jun Ma Æ Ya Jia Received: 6 April 2009 / Revised: 8 June 2009 / Accepted: 10 June 2009 / Published online: 7 July 2009 Ó European Biophysical Societies’ Association 2009 Abstract It is well known that the spatial distribution of the calcium ion channels in the endoplasmic reticulum is dis- crete. We study the Ca 2? spiral pattern formation based on a model in which ion channels are discretely and randomly distributed. Numerical simulations are performed on dif- ferent types of media with the Ca 2? release sites uniformly distributed, discretely and uniformly arranged, or discretely and randomly arranged. The comparisons among the dif- ferent media show that random distribution is necessary for spontaneous initiation of Ca 2? spiral waves, and the discrete and random distribution is of significance for spiral waves under physiologically reasonable conditions. The period and velocity of spiral waves are also calculated, and they are not prominently changed by varying the type of medium. Keywords Ca 2? spiral waves  Ion channel  Random medium Introduction Ca 2? is used as a second messenger in cell signalling. Most of the calcium ions that constitute the signal are released from intracellular stores, e.g., the endoplasmic reticulum (ER) or sarcoplasmic reticulum (SR), in which the normal Ca 2? concentration is much higher than that of cytosol inside the intracellular space. Ca 2? is released from intra- cellular stores through two types of ion channels: the ryanodine receptor (RyR) and the inositol 1,4,5-trisphos- phate receptor (IP 3 R). Ca 2? cannot transmit information by its binding specificity or simply by its presence. Conse- quently, the signal is encoded in temporal and spatial patterns, such as Ca 2? waves (Falcke 2004). Intracellular calcium waves were first observed in medaka eggs (Ridg- way et al. 1977) and then found in Xenopus oocytes and other cell types (Sanderson et al. 1994; Robb-Gaspers and Thomas 1995; Thomas et al. 1995; Harris-White et al. 1998; Lipp and Niggli 1993). In Xenopus oocytes, IP 3 R is the only type of ion channel in the ER. It is widely accepted that the release channels are spatially organized in clusters, and there are only a few tens of intact IP 3 receptor channels in square-micrometer- sized clusters (Bootman et al. 1997; Foskettet et al. 2007). Many effects of this type of discrete distribution of ion channels in the ER have been experimentally studied. The most prominent one is the transition from local calcium release to calcium wave propagation (Parker and Yao 1991; Yao et al. 1995; Parker and Yao 1996; Yagodin et al. 1995, 1994; Cheng et al. 1996). For example, in Xenopus oocytes, with low IP 3 concentration, Ca 2? puffs— Ca 2? releasing from the serval ion channel in a cluster— are observed. As IP 3 concentration increases above a threshold value, a Ca 2? wave is able to propagate from the site of the initial Ca 2? release (Parker and Yao 1991, 1996; Yao et al. 1995). Furthermore, it is known that the discrete release sites are randomly arranged in the ER. To our knowledge, the effects of the random distribution of Ca 2? release sites have not been extensively studied. Bugrim J. Tang (&) College of Science, China University of Mining and Technology, 221008 Xuzhou, China e-mail: tjuns1979@yahoo.com.cn J. Tang  L. Yang  J. Ma  Y. Jia Department of Physics and Institute of Biophysics, Huazhong Normal University, 430079 Wuhan, China J. Ma Department of Physics, Lanzhou University of Technology, 730050 Lanzhou, China 123 Eur Biophys J (2009) 38:1061–1068 DOI 10.1007/s00249-009-0509-y