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Transition from Stochastic to Deterministic Behavior in Calcium Oscillations

Ursula Kummer ^*, Borut Krajnc , Jürgen Pahle ^*, Anne K. Green , C. Jane Dixon  and Marko Marhl 

^* Bioinformatics and Computational Biochemistry Group, EML Research, D-69118 Heidelberg, Germany; Department of Physics, Faculty of Education, University of Maribor, SI-2000 Maribor, Slovenia; Department of Biological Sciences, The University of Warwick, Coventry, CV4 7AL, United Kingdom; and Leicester School of Pharmacy, De Montfort University, Leicester, LE1 7BH, United Kingdom

Correspondence: Address reprint requests to Dr. Ursula Kummer, Tel.: 49-6221-533225; Fax: 49-6221-533298; E-mail: ursula.kummer{at}eml-r.villa-bosch.de.

Simulation and modeling is becoming more and more important when studying complex biochemical systems. Most often, ordinary differential equations are employed for this purpose. However, these are only applicable when the numbers of participating molecules in the biochemical systems are large enough to be treated as concentrations. For smaller systems, stochastic simulations on discrete particle basis are more accurate. Unfortunately, there are no general rules for determining which method should be employed for exactly which problem to get the most realistic result. Therefore, we study the transition from stochastic to deterministic behavior in a widely studied system, namely the signal transduction via calcium, especially calcium oscillations. We observe that the transition occurs within a range of particle numbers, which roughly corresponds to the number of receptors and channels in the cell, and depends heavily on the attractive properties of the phase space of the respective systems dynamics. We conclude that the attractive properties of a system, expressed, e.g., by the divergence of the system, are a good measure for determining which simulation algorithm is appropriate in terms of speed and realism.

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